^°l' 


IN  MEMORIAM 
FLOR1AN  CAJOR1 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/advancedarithmetOOwentrich 


AN 


ADVANCED  ARITHMETIC 


HIGH  SCHOOLS,  NORMAL  SCHOOLS 
AND   ACADEMIES 


BY 


G.  A.  WENTWORTH,  A.M. 

AUTHOR  OF  A  SERIES  OF  TEXT-BOOKS  IN  MATHEMATICS 


BOSTON,  U.S.A. 

GINN   &  COMPANY,   PUBLISHERS 

(ZDbe  &tljeu£ttm  JJteaa 

1898 


Copyright,  1898,  by 
GEORGE  A.  WENTWORTH 


AJAj  rights  reserved 


PREFACE 


Every  high  school,  normal  school,  and  academy  should 
allow  sufficient  time  for  a  thorough  review  of  Arithmetic. 
This  book  has  been  written  as  a  text-book  for  that  purpose, 
and  for  that  purpose  only.  It  is  not  intended  for  begin- 
ners, but  assumes  that  pupils  have  previously  read  a  more 
elementary  Arithmetic. 

The  shortest  and  surest  road  to  a  knowledge  of  Arith- 
metic is  by  solving  problems.  This  work  is  abundantly 
supplied  with  welbgraded,  practical  problems,  many  taken 
from  Wentworth  and  Hill's  High  School  Arithmetic,  but 
many  of  them  are  new,  and  of  a  kind  to  meet  the  require- 
ments of  the  present  time.  These  problems  are  designed 
to  convey  a  great  amount  of  useful  information,  as  well  as 
to  furnish  the  very  best  mental  training,  the  primary  object 
of  the  study.  They  cover  to  a  great  extent  the  field  of 
mercantile  transactions,  and  so  far  as  practicable  the  field 
of  science.  It  is  not  necessary  for  any  pupil,  or  any  class 
even,  to  do  all  the  problems.  Every  teacher  Can  select 
such  chapters  and  such  parts  of  chapters  as  are  suited  to 
the  needs  of  his  pupils. 

Decimals  are  introduced  at  the  beginning  of  the  book. 
Numbers  on  each  side  of  the  decimal  point  perform  pre- 
cisely the  same  office.  The  only  difference  is  that  numbers 
at  the  left  of  the  decimal  point  count  ivhole  units,  and 
numbers  at  the  right  count  equal  parts  of  the  unit.  Pupils 
learn  the  notation  on  both  sides  of  the  decimal  point  as 


M30GO73 


IV  PREFACE. 

easily  as  on  one  side,  provided  they  have  a  clear  conception 
of  the  units  counted.  Dimes  and  cents  are  good  examples 
of  tenths  and  hundredths  of  a  dollar ;  but  decimeters,  centi- 
meters, and  millimeters,  marked  on  a  meter  stick,  are  the 
best  examples  of  tenths,  hundredths,  and  thousandths  of  a 
unit,  in  general.  The  Metric  System  is  taught  naturally 
in  connection  with  decimals,  and  is  easily  learned.  Only 
the  units  employed  furnish  any  difficulty.  The  great 
number  of  problems  given  under  the  Metric  System  is 
to  familiarize  the  learner  with  the  units  of  the  system, 
to  show  the  simplicity  of  the  system  in  its  application  to 
everyday  problems,  and  at  the  same  time  to  give  practice 
in  operations  involving  decimals.  This  system  is  used  in 
the  laboratories  of  science,  and  in  international  transac- 
tions. Though  not  yet  adopted  by  the  United  States  in 
the  common  affairs  of  life,  it  has  certainly  forced  its 
way  to  a  position  requiring  recognition  in  all  secondary 
schools  of  the  country. 

The  introduction  of  •  logarithms  into  the  High  School 
Arithmetic  was  warmly  welcomed  by  progressive  teachers ; 
and  the  chapter  on  that  subject  in  this  book  has  been 
written  with  special  reference  to  acquiring  easily  the 
practical  use  of  a  four-place  table. 

Every  effort  has  been  made  to  avoid  errors  in  problems 

and  answers.     The  author  will  be  very  grateful  to  any  one 

who  will  call  his  attention  to  any  mistake  that  may  be 

discovered. 

G.  A.  WENTWORTH. 

Exeter,  N.  H.,  June,  1898. 


CONTENTS. 


CHAPTER  PAGE 

I.  Notation  and  Numeration         .....  1 

II.  Addition  and  Subtraction 12 

III.  Multiplication 24 

IV.  Division 35 

V.  Metric  Measures 58 

VI.  Measures  and  Multiples  of  Numbers          .         .  94 

VII.  Common  Fractions 109 

VIII.  Compound  Quantities      ......  147 

IX.  Problems .179 

X.  Metric  and  Common  Systems         ....  210 

XL  Ratio  and  Proportion 217 

XII.  Percentage 237 

XIII.  Interest  and  Discount 259 

XIV.  Stocks  and  Bonds,  Exchange,  Accounts                .  288 
XV.  Powers  and  Roots .  306 

XVI.  Mensuration 317 

XVII.  Continued  Fractions  and  Scales  of  Notation         .  332 

XVIII.  Series 338 

XIX.  Common  Logarithms   .......  345 

XX.  Applications  of  Logarithms           ....  359 

XXI.  Miscellaneous  Problems 372 


VOCABULARY. 


Abstract  numbers.  Numbers  used  without  reference  to  any  par- 
ticular unit,  as  8,  10,  21.  All  numbers  are  in  themselves  abstract 
whether  the  kind  of  thing  numbered  is  mentionedor  is  not  meat 

Addition.     The  process  of  combining  two  or  more  numbers. 

Agent.     A  person  who  transacts  business  for  another. 

Aliquot  parts  of  a  given  number.  Numbers  that  are  contained  an 
integral  number  of  times  in  the  given  number. 

Alligation.  The  process  of  finding  the  value  of  a  mixture  of  quanti- 
ties of  different  values;  or  of  finding  the  proportion  of  quantities  of 
different  values  to  be  used  to  make  a  mixture  of  a  given  value. 

Amount.  The  sum  of  two  or  more  numbers.  In  interest,  the  sum 
of  the  principal  and  interest. 

Ampere.  The  unit  of  quantity  of  electricity.  The  current  produced 
by  a  force  of  one  volt  in  a  circuit  of  a  resistance  of  one  ohm. 

Analysis.  The  process  of  reasoning  from  the  given  number  to  one, 
and  then  from  one  to  the  required  number. 

Annual  interest.  Simple  interest  on  the  principal  and  on  each  year's 
interest  from  the  time  each  interest  is  due  until  settlement. 

Annuity.     A  sum  of  money  to  be  paid  at  regular  intervals  of  time. 

Antecedent.     The  first  term  of  a  ratio. 

Antilogarithm.     The  number  corresponding  to  a  logarithm. 

Area  of  a  surface.     The  number  of  square  units  in  the  surface. 

Arithmetic.  The  science  that  treats  of  numbers  and  the  methods  of 
using  them. 

Arithmetical  progression.  A  series  of  numbers  that  increase  or 
decrease  by  a  common  difference. 

Assets.     All  the  property  of  an  estate,  individual,  or  corporation. 

Average  of  numbers.  The  number  that  can  be  put  for  each  of  the 
numbers  without  altering  their  sum. 

Average  of  payments.  The  average  time  at  which  several  payments 
due  at  different  dates  may  be  equitably  made. 


VOCABULARY.  Vll 

Bank.     An  establishment  for  the  custody,  loaning,  and  exchange  of 

money  ;  and  often  for  the  issue  of  money. 
Bank  discount.     An  allowance  received  by  a  bank  for  the  loan  of 

money,  paid  as  interest  at  the  time  of  lending. 
Bills.    Written  statements  of  goods  sold,  or  services  rendered,  giving 

the  price  and  date  of  each  item,  and  the  parties  concerned. 
Bonds.     Written  contracts  under  seal  to  pay  specified  sums  of  money 

at  specified  times,  issued  by  national  governments,  states,  cities, 

and  other  corporations. 
Breakage.     In  customs,  an  allowance  of  a  certain  per  cent  on  liquors 

in  bottles  ;  also  on  glassware  and  china. 
Broker.     A  person  who  buys  and  sells  stocks  and  bonds  for  another. 
Brokerage.     The  commission  charged  by  a  broker. 
Cancellation.     The  striking  out  of  a  common  factor  from  the  divi- 
dend and  divisor. 
Centrifugal  force.     The  force  that  tends  to  make  a  revolving  body 

move  in  a  straight  line. 
Characteristic  of  a  logarithm.     The  integral  part  of  the  logarithm. 
Check.     A  draft  upon  a  bank  where  the  maker  has  money  deposited. 
Cologarithm  of  a  number.     The  logarithm  of  the  reciprocal  of  the 

number. 
Commercial  discount.     A  reduction  from  the  list  price  of  an  article, 

from  the  amount  of  a  bill,  or  from  the  amount  of  a  debt. 
Commission.    Compensation  for  the  transaction  of  business,  reckoned 

at  some  per  cent  of  the  money  employed  in  the  transaction. 
Common  denominator.     A  denominator  common  to  two  or  more 

fractions. 
Common  factor.     A  factor  common  to  two  or  more  numbers. 
Common  fractions.     Fractions   expressed   by   two   numbers,    one 

under  the  other,  with  a  line  between  them. 
Common  measure  of  two  or  more  numbers.      A   number  that 

exactly  divides  each  of  them. 
Common  multiple  of  two  or  more  numbers.     A  number  exactly 

divisible  by  each  of  them. 
Complex  fraction.     A  fraction  that  has  a  fraction  in  one  or  both  of 

its  terms. 
Composite  number.     The  product  of  two  or  more  integral  factors, 

each  factor  being  greater  than  unity. 
Compound  fraction.     A  fraction  of  an  integer  or  of  a  fraction. 
Compound  interest.     Interest  not  paid  when  due,  but  added  to  the 

principal  at  regular  intervals. 


Vlll  VOCABULARY. 

Compound   quantities.     Quantities  expressed  in  two  or  more  de- 
nominations. 
Concrete  numbers.     Numbers  applied  to  specified  things. 
Consequent.     The  second  term  of  a  ratio. 
Consignee.     The  person  or  firm  to  whom  goods  are  sent. 
Consignor.     The  person  or  firm  who  sends  goods  to  another. 
Continued  fraction.     A  fraction  whose  numerator  is  1,  and  whose 

denominator  is  an    integer    plus    a    fraction   whose    numerator 

is  1,  and  whose  denominator  is  an  integer  plus  a  fraction,  and 

so  on. 
Cooperative  bank.     A  mutual  corporation  for  the  accumulation  of 

a  capital  to  be  loaned  to  its  members. 
Corporation.     An  association  of  individuals  authorized  by  law  to 

transact  business  as  a  single  person. 
Couplet     The  two  terms  of  a  ratio  taken  together. 
Coupon.     A  certificate  of  interest  attached  to  a  bond,  to  be  cut  off 

when  due  and  presented  for  payment. 
Creditor.     A  person  or  firm  to  whom  money  is  due. 
Cube  root  of  a  number.     One  of  its  three  equal  factors. 
Currency.     The  medium  of  exchange  I  mployed  in  business. 
Customs.     Duties  or  taxes  imposed  by  law  on  merchandise  imported, 

and  sometimes  on  merchandise  exported. 
Debtor.     A  person  or  firm  who  owes  money  to  another. 
Decimal  point.     A  dot  placed  between  the  number  that  counts  whole 

units  and  the  number  that  counts  decimal  units. 
Decimals.     Fractions  of  which  only  the  numerators  are  written,  and 

the  denominators  are  ten  or  some  power  of  ten. 
Decimal  system.     The  common  system  of  numbers  founded  on  their 

relations  to  ten. 
Demand  notes.     Notes  that  are  payable  on  demand. 
Denominator.     The  number  that  shows  into  how  many  equal  parts 

a  unit  is  divided. 
Difference.     The  number  found  by  subtraction. 
Discount.     An  allowance  made  for  the  payment  of  money  before  it 

becomes  due.     Also,  the  difference  between  the  market  value  and 

the  face  value  when  the  market  value  is  below  the  face  value. 
Dividend.     In  division,  the  number  to  be  divided.     In  business,  the 

sum  paid  on  each  share  of  stock  from  the  profits  of  the  business. 
Division.     The  operation  of  finding  the  other  factor,  when  a  product 

and  one  of  its  factors  are  given. 
Divisor.     The  number  by  which  a  given  dividend  is  to  be  divided. 


VOCABULARY.  IX 

Draft.     A  written  order  directing  one  person  or  firm  to  pay  a  specified 

sum  of  money  to  another. 
Drawee  of  a  draft.     The  person  on  whom  the  draft  is  drawn. 
Drawer  of  a  draft.     The  person  who  signs  the  draft. 
Duties.     Taxes  required  by  the  government  to  be  paid  on  goods  im- 
ported, exported,  or  put  on  the  market  for  consumption. 
Endorser  of  a  note.     A  person  who  writes  his  name  on  the  back  of 

the  note.     The  endorser  is  responsible  for  the  payment  of  the  note 

unless  he  writes  above  his  name  the  words  without  recourse. 
Equation.     A  statement  that  two  expressions  of  number  are  equal. 
Equivalents.     Equals  in  value,  area,  or  volume. 
Even  numbers.     Numbers  exactly  divisible  by  2. 
Evolution.     The  process  of  finding  the  root  of  a  number. 
Exact  interest.     Simple  interest  reckoning  365  days  to  the  year. 
Example.     A  question  to  be  solved. 
Exchange.     A  system  of  paying  debts,  due  to  persons  living  at  a 

distance,  by  transmitting  drafts  instead  of  money. 
Exponent.     A  small  figure  placed  at  the  right  of  a  number  to  show 

how  many  times  the  number  is  taken  as  a  factor. 
Extremes.     The  first  and  last  terms  of  a  proportion. 
Face  of  a  note  or  draft.     The  sum  of  money  named  in  it. 
Factors.     A  set  of  numbers  whose  product  is  the  given  number. 

In  commerce,  agents  employed  by  merchants  to  transact  business. 
Figures.     Symbols  used  to  represent  numbers  in  the  Arabic  system. 

Also  diagrams  used  to  represent  geometrical  forms. 
Firm.     The  name  under  which  a  company  transacts  business. 
Foot-pound.     The  unit  of  work.     The  work  done  in  raising  a  weight 

of  one  pound  to  a  height  of  one  foot. 
Force.     That  which  tends  to  produce  motion. 
Fractions.     One  or  more  of  the  equal  parts  of  a  unit. 
Fulcrum.     The  point  or  line  on  which  a  lever  turns. 
Gain.     The  selling  price  minus  the  cost  price. 
Geometrical  progression.      A  series   of  numbers,    each  term  of 

which  after  the  first  is  obtained  by  multiplying  the  preceding  term 

by  a  constant  multiplier. 
Grace.     An  allowance  of  three  days,  after  the  date  a  note  becomes 

due,  within  which  to  pay  the  note. 
Gram.     The  unit  of  weight  in  the  metric  system. 
Gravitation.     The  force  by  which  all  bodies  attract  each  other. 
Greatest  common  measure   of  two   or  more   numbers.      The 

greatest  number  that  will  exactly  divide  each  of  them. 


X  VOCABULARY. 

Harmonical  progression.  A  series  of  numbers,  the  reciprocals  of 
which  form  an  arithmetical  progression. 

Holder  of  a  note.     The  person  who  has  legal  possession  of  it. 

Horse  power.    The  power  to  do  33,000  foot-pounds  of  work  a  minute. 

Improper  fraction.  A  fraction  whose  numerator  equals  or  exceeds 
its  denominator. 

Index.  A  figure  written  at  the  left  and  above  the  radical  sign  to 
show  what  root  of  the  number  under  the  radical  sign  is  required. 
A  fraction  written  at  the  right  of  a  number,  of  which  the  numer- 
ator shows  the  required  power  of  that  number,  and  the  denomi- 
nator the  required  root  of  that  power. 

Instalment.  A  payment  in  part- 
Insurance.  A  guarantee  by  an  insurance  company  of  a  specified  sum 
of  money  to  the  person  insured  in  the  event  of  loss  of  property  by 
fire,  by  storm  at  sea,  or  by  other  specified  disaster ;  or  in  the  event 
of  the  death  of  the  person  insured,  or  of  accident  to  him. 

Integral  numbers.     Numbers  that  denote  whole  units. 

Interest.     Money  paid  for  the  use  of  money. 

Involution.     The  process  of  finding  a  power  of  a  number. 

Latitude  of  a  place.  The  distance  north  or  south  from  the  equator, 
expressed  in  degree  measures. 

Leakage.  In  customs,  an  allowance  of  a  certain  per  cent  on  liquors 
in  barrels  or  casks. 

Least  common  denominator  of  two  or  more  fractions.  The  least 
common  multiple  of  their  denominators. 

Least  common  multiple  of  two  or  more  numbers.  The  least 
number  that  is  exactly  divisible  by  each  of  them. 

Lever.     A  rigid  bar  that  will  move  freely  about  its  fulcrum. 

Liability.     A  debt,  or  obligation  to  pay. 

Like  numbers.     Numbers  applied  to  the  same  unit. 

Line.     Length  without  breadth  or  thickness. 

Liter.     The  unit  of  capacity  in  the  metric  system. 

Logarithm  of  a  number.  The  exponent  of  the  power  to  which  the 
base  must  be  raised  to  obtain  the  given  number. 

Long  division.  The  method  of  dividing  in  which  the  processes  are 
written  in  full. 

Longitude  of  a  place.  The  distance  east  or  west  from  a  standard 
meridian,  expressed  in  degree  measures. 

Loss.     The  cost  price  minus  the  selling  price. 

Mantissa  of  a  logarithm.     The  decimal  part  of  the  logarithm. 

Maturity  of  a  note.     The  date  a  note  legally  becomes  due. 


VOCABULARY.  XI 

Mean  proportional.     A  number  that  is  both  the  second  and  third 

terms  of  a  proportion. 
Means.     The  terms  of  a  proportion  between  the  extremes. 
Measures  of  a  number.     The  exact  divisors  of  the  numbers. 
Meridians.   Imaginary  lines  drawn  straight  around  the  earth  through 

both  poles. 
Meter.     The  unit  of  length  in  the  metric  system. 
Metric  system.     A  system  of  weights  and  measures  expressed  in 

the  decimal  scale. 
Minuend.     The  number  from  which  the  subtrahend  is  taken. 
Mixed  number.     A  whole  number  and  a  fraction. 
Money  of  a  country.     The  legal  currency  of  the  country. 
Multiple  of  a  number.     The  product  obtained  by  taking  the  given 

number  an  integral  number  of  times. 
Multiplicand.     The  number  to  be  multiplied  by  another. 
Multiplication.     The  operation  of  taking  a  number  of  units  a  num- 
ber of  times. 
Multiplier.     The  number  by  which  the  multiplicand  is  multiplied. 
Net  proceeds.     The  money  that  remains  of  the  money  received  for 

property  after  all  expenses  and  discounts  are  paid. 
Notation.     A  system  of  expressing  numbers  by  symbols. 
Note.     A  written  agreement  to  pay,  for  value  received,  a  specified 

sum  of  money  on  demand  or  at  a  specified  time. 
Numbers.     Expressions  applied  to  a  unit  to  show  how  many  times 

the  unit  is  taken. 
Numeration.     A  system  of  naming  numbers. 
Numerator.     The  number  that  shows  how  many  are  taken  of  the 

equal  parts  into  which  a  unit  is  divided. 
Obligation.     A  debt,  or  liability  to  pay. 
Odd  numbers.     Numbers  not  exactly  divisible  by  2. 
Ohm.     The  unit  of  resistance  to  electricity.     The  resistance  of  a 

column  of  mercury,  l<imm  in  cross  section  and  106cm  long  at  0°  C. 
Order  of  units.     The  number  of  things  in  a  group,  as  tens,  hundreds. 
Partial  payment.     Part  payment  on  a  note. 
Partnership.     An  association  of  persons  to  carry  on  business. 
Par  value.     Face  or  nominal  value. 
Pendulum.     A  body  suspended  by  a  straight  line  from  a  fixed  point, 

and  moving  freely  about  that  point  as  a  centre. 
Percentage  of  a  number.     One  or  more  hundredths  of  the  number. 
Perimeter.     The  length  of  the  boundary  of  a  plane  figure. 
Period.     A  group  of  three  figures. 


Xll  VOCABULARY. 

Policy.     A  written  contract  of  insurance. 

Poll  tax.     A  tax  levied  by  the  head  or  poll. 

Power.     The  product  of  two  or  more  equal  factors. 

Premium.  Money  paid  for  insurance.  Also  the  excess  of  market 
value  above  par  value. 

Present  worth.     The  present  value  of  a  debt  due  at  a  future  time. 

Prime  number.  A  number  that  cannot  be  exactly  divided  by  any 
number  except  itself  and  one. 

Principal.     Money  drawing  interest. 

Problem.     A  question  to  be  solved. 

Proceeds  or  avails  of  a  note  or  draft  The  amount  of  the  note  or 
draft  less  the  discount  and  exchange. 

Product.     The  Dumber  found  by  multiplication. 

Proof.     The  evidence  that  establishes  the  accuracy  of  any  result. 

Proper  fraction.  A  fraction  whose  numerator  is  less  than  its  de- 
nominator. 

Proportion.     A  statement  that  two  ratios  are  equal. 

Protest.  A  notice  in  writing  by  a  notary  public  to  the  endorsers  of 
a  note  that  it  has  not  been  paid. 

Pulley.  A  grooved  wheel  that  turns  freely  within  a  block,  fixed  or 
movable,  by  means  of  a  rope  that  passes  over  the  groove. 

Quotient.     The  number  found  by  division. 

Rate  per  cent.     Rate  by  the  hundred. 

Ratio.  The  relative  magnitude  of  two  numbers  or  of  two  quantities, 
when  expressed  by  the  quotient  of  the  first  divided  by  the  second. 

Receipt.     A  written  acknowledgment  of  money  or  goods  received. 

Reciprocal  of  a  number.     One  divided  by  thit  number. 

Reduction.  The  process  of  changing  the  unit  in  which  a  quantity  is 
expressed  without  changing  the  value  of  the  quantity. 

Remainder.     The  number  found  by  subtraction. 

Repeating  or  circulating  decimals.  Decimals  that  contain  a  con- 
stantly recurring  figure  or  series  of  figures. 

Repetend.  The  figure  or  series  of  figures  that  constantly  recurs  in  a 
repeating  decimal. 

Root  of  a  number.     One  of  the  equal  factors  of  the  number. 

Rule.     The  statement  of  a  prescribed  method. 

Savings  bank.  A  bank  to  receive  deposits  and  pay  compound  inter- 
est to  the  depositors. 

Screw.  A  cylinder  that  has  on  its  surface  a  uniform  projection  in 
the  form  of  a  spiral  curve,  called  the  thread. 

Security.     Property  used  to  guarantee  the  payment  of  any  debt. 


VOCABULARY.  Xlll 

Share.     One  of  a  certain  number  of  equal  parts  into  which  the  capi- 
tal of  a  company  is  divided. 
Short  division.     The  method  of  dividing  in  which  the  operations  of 

multiplying  and  subtracting  are  performed  mentally. 
Similar  fractions.     Fractions  that  have  a  common  denominator. 
Simple  fractions.     Fractions  whose  terms  are  integral  numbers. 
Simple  quantities.     Quantities  expressed  in  a  single  denomination. 
Sinking  fund.     The  final  value  of  sums  of  money  put  at  interest  at 

regular  intervals  of  time,  to  pay  a  debt  due  at  a  stated  time. 
Solid.     A  magnitude  that  has  length,  breadth,  and  thickness. 
Solution.    The  process  by  which  the  answer  to  a  question  is  obtained. 
Specific  gravity  of  a  substance.     The  ratio  of  the  weight   of   a 

given  volume  of  it  to  the  weight  of  an  equal  volume  of  water. 
Square  root  of  a  number.     One  of  its  two  equal  factors. 
Stock.     Capital  invested  in  business. 
Stock  company.     An  association  of  persons  under  the  laws  of  the 

state  for  the  purpose  of  carrying  on  a  specified  business. 
Subtraction.     The  process  of  taking  one  number  from  another. 
Subtrahend.     The  number  that  is  taken  from  the  minuend. 
Sum.     The  number  found  by  addition. 
Surd.     An  indicated  root,  that  cannot  be  exactly  found. 
Surface.     A  magnitude  that  has  length  and  breadth. 
Tare.     In  customs,  an  allowance  for  the  weight  of  the  box,  cask,  bag, 

or  other  wrapping. 
Taxes.     Money  required  of  persons  and  corporations  for  the  support 

of  the  government  and  for  other  public  purposes. 
Thermometer.     An  instrument  for  measuring  temperature. 
Time  notes.     Notes  that  are  payable  at  a  specified  time. 
Units.      The    standards  by  which   we   count    separate    objects   or 

measure  magnitudes. 
Velocity.     Rate  of  motion,  measured  by  the  number  of  units  of 

space  passed  over  in  a  unit  of  time. 
Verify.     To  establish  by  trial  the  truth  of  any  statement. 
Volt.     The  unit  of  force  of  electricity.    The  force  required  to  send  one 

ampere  of  electricity  through  a  circuit  of  a  resistance  of  one  ohm. 
Volume  of  a  solid.     The  number  of  cubic  units  in  the  solid. 
Wheel  and  axle.     A  simple  machine,  consisting  of  a  wheel  firmly 

attached  to  an  axle. 
Work.     The  act  of  changing  the  position  of  a  body  by  overcoming 

resistance  to  the  change. 
Yard*     The  unit  of  length  in  the  common  system, 


XIV  SHORT   PROCESSES. 


Short  Processes. 

Note.  These  processes  should  be  learned  as  fast  as  they  can  be  utilized 
in  the  ordinary  work  of  Arithmetic.  The  time  to  learn  each  process  is  left 
to  the  discretion  of  the  teacher. 

1.  To  multiply  by  25  (±  of  100), 

Multiply  by  100  and  divide  the  product  by  %. 

2.  To  divide  by  25, 

Multiply  by  4  and  divide  the  product  by  100. 

3.  To  multiply  by  2.5  or  2£  (\  of  10), 
Multiply  by  10  and  divide  the  product  by  4,. 

4.  To  divide  by  2.5  or  2£, 

Multiply  by  4  and  divide  the  product  by  10. 

5.  To  multiply  by  50  (£  of  100), 

Multiply  by  100  and  divide  the  product  by  2. 

6.  To  divide  by  50, 

Multiply  by  2  and  divide  the  product  by  100. 

7.  To  multiply  by  75  (f  of  100), 

Multiply  by  100  and  subtract  from  the  product  \  of  it. 

8.  To  divide  by  75, 

Divide  by  100  and  add  to  the  quotient  £  of  it. 

9.  To  multiply  by  33£  (£  of  100), 
Multiply  by  100  and  divide  the  product  by  S. 

10.  To  divide  by  33£, 

Multiply  by  3  and  divide  the  product  by  100. 

11.  To  multiply  by  3£  (£  of  10), 

Multiply  by  10  and  divide  the  product  by  S. 

12.  To  divide  by  3£, 

Multiply  by  8  and  divide  the  product  by  10. 

13.  To  multiply  by  833$  (£  of  1000), 
Multiply  hy  1000  and  divide  the  product  by  8. 


SHORT    PROCESSES.  XV 

14.  To  divide  by  333£, 

Multiply  by  3  and  divide  the  product  by  1000. 

15.  To  multiply  by  16f  Q  of  100), 
Multiply  by  100  and  divide  the  product  by  6. 

16.  To  divide  by  16f, 

Multiply  by  6  and  divide  the  product  by  100. 

17.  To  multiply  by  166f  (£  of  1000), 
Multiply  by  1000  and  divide  the  product  by  6. 

18.  To  divide  by  166§, 

Multiply  by  6  and  divide  the  product  by  1000. 

19.  To  multiply  by  66f  (f  of  100), 

Multiply  by  100  and  subtract  from  the  product  £  of  it. 

20.  To  divide  by  66f, 

Divide  by  100  and  add  to  the  quotient  £  of  it. 

21.  To  multiply  by  12£  (£  of  100), 
Multiply  by  100  and  divide  the  product  by  8. 

22.  To  divide  by  12£, 

Multiply  by  8  and  divide  the  product  by  100. 

23.  To  multiply  by  14f  (i  of  100), 
Multiply  by  100  and  divide  the  product  by  7. 

24.  To  divide  by  14f, 

Multiply  by  7  and  divide  the  product  by  100. 

25.  To  multiply  by  a  number  that  is  a  little  less  than  some 

multiple  of  10,  as  100,  1000,  etc., 

Multiply  the  multiplicand  by  the  multiple  of  10  that  differs 
little  from  the  given  multiplier.  Then  multiply  the  multipli- 
cand by  the  difference  between  this  multiple  of  10  and  the 
given  multiplier,  and  find  the  difference  of  the  two  products. 

Thus,  to  multiply  by  998  (1000  —  2),  multiply  by  1000  and  then  by  2,  and 
take  the  difference  of  the  products. 


NOTICE    TO    TEACHERS. 


Pamphlets  containing  the  answers  will  be  furnished 
without  charge  to  teachers  for  their  classes,  on  appli- 
cation to  GrINN  £  COMPANY',  Publishers. 


ADVANCED    ARITHMETIC. 


CHAPTER  I. 

NOTATION  AND  NUMERATION. 

1.  Units.  The  standards  by  which  we  count  or  measure 
are  called  units. 

The  unit  may  be  a  single  thing  or  a  definite  group  of  things.  Thus, 
in  counting  the  eggs  in  a  nest  the  unit  is  an  egg ;  in  selling  eggs  by 
the  dozen  the  unit  is  a  dozen  eggs;  in  selling  bricks  by  the  thousand 
the  unit  is  a  thousand  bricks;  in  measuring  short  distances  the  unit  is 
an  inch,  aioot,  or  a  yard;  in  measuring  long  distances  the  unit  is  a 
rod  or  a  mile. 

2.  Numbers.  Expressions  applied  to  a  unit  to  show 
how  many  times  the  unit  is  taken  are  called  numbers. 

Thus,  if  we  put  an  apple  into  an  empty  fruit  dish,  then  another, 
and  then  another,  we  shall  have  three  apples  in  the  dish.  Here  an 
apple  is  the  unit,  and  three  is  the  number  of  times  the  unit  is  taken. 

3.  Integral  Numbers.  Numbers  applied  to  whole  units 
are  called  whole  numbers,  integral  numbers,  or  integers. 

4.  Figures.  The  following  symbols,  called  figures,  or 
digits,  are  used  to  represent  the  numbers  of  Arithmetic  : 

012  3  4567  8  9 

Zero    One    Two    Three    Four    Five    Six     Seven    Eight    Nine 

The  first  figure,  0,  is  called  zero,  naught,  or  cipher,  and 
stands  for  the  words  no  number.  Each  of  the  other  figures 
stands  for  the  number  whose  name  is  written  below  it. 


2  NOTATION    AND   NUMERATION. 

5.  Tens.  The  next  number,  ten,  is  expressed  by  writing 
0  at  the  right  of  1.  Thus,  ten  is  written  10.  In  this  posi- 
tion 1  signifies  not  one,  but  one  group  of  ten  ones. 

Figures  signifying  tens  are  written  in  the  second  place 
from  the  right. 

In  the  same  way  twenty  (2  tens)  is  expressed  by  20  ; 
thirty  (3  tens)  by  30 ;  forty  (4  tens)  by  40  ;  fifty  (5  tens) 
by  50  ;  sixty  (6  tens)  by  60 ;  seventy  (7  tens)  by  70  ; 
eighty  (8  tens)  by  80  ;  ninety  (9  tens)  by  90. 


6.  Tens  and  Ones.  A  number  containing  tens  and  ones 
is  expressed  by  writing  the  figure  for  the  tens  in  the 
second  place  from  the  right,  and  the  figure  for  the  ones  in 
the  first  place. 


Eleven, 

one  ten   and  one, 

is  expressed  by  11. 

Twelve, 

one  ten   and  two, 

is  expressed  by  12. 

Thirteen, 

one  ten   and  three, 

is  expressed  by  13. 

Fourteen, 

one  ten   and  four, 

is  expressed  by  14. 

Fifteen, 

one  ten   and  five, 

is  expressed  by  15. 

Sixteen, 

one  ten   and  six, 

is  expressed  by  16. 

Seventeen, 

one  ten   and  seven, 

is  expressed  by  17. 

Eighteen, 

one  ten   and  eight, 

is  expressed  by  18. 

Nineteen, 

one  ten   and  nine, 

is  expressed  by  19. 

Twenty-one, 

two  tens  and  one, 

is  expressed  by  21. 

Twenty-two, 

two  tens  and  two, 

is  expressed  by  22. 

Forty-three, 

four  tens  and  three, 

is  expressed  by  43. 

Fifty-four, 

five  tens  and  four, 

is  expressed  by  54. 

Sixty-five, 

six    tens  and  five, 

is  expressed  by  65. 

And  so  on  to  ninety-nine  (nine  tens  and  nine)  which  is 
by  99. 


7.  Hundreds.  A  group  of  10  tens  is  called  a  hundred, 
and  figures  signifying  hundreds  are  written  in  the  third 
place  from  the  right. 

Thus,  one  hundred,  two  hundreds,  three  hundreds,  etc.,  are  ex- 
pressed by  100,  200,  300,  etc. 


NOTATION    AND    NUMERATION.  3 

8.  To  Write  Hundreds,  Tens,  and  Ones.  We  write  first 
the  hundreds,  then  the  tens  and  ones. 

Thus,  two  hundred  seventy-six  is  written  276. 

Express  in  figures  the  following  numbers  : 
Seven  hundred  sixty-five.  Nine  hundred. 

One  hundred  twenty-three.         Five  hundred  eighty-one. 
Six  hundred  ninety-four.  Four  hundred  thirty. 

Nine  hundred  forty-six.  Seven  hundred  nine. 

Two  hundred  twenty-nine.  Seven  hundred  ninety. 

One  hundred  ten.  Seven  hundred  ninety-nine. 

9.  To  Read  Hundreds,  Tens,  and  Ones.  We  read  first 
the  hundreds,  then  the  tens  and  ones. 

Thus,  the  number  359  has  3  hundreds,  5  tens,  and  9  ones,  and  is 
read  three  hundred  fifty-nine;  the  number  807  has  8  hundreds,  no 
tens,  and  7  ones,  and  is  read  eight  hundred  seven. 

Note.  In  reading  359,  807,  or  any  other  integral  number,  do  not 
introduce  the  word  and ;  that  is,  do  not  say  three  hundred  and  fifty- 
nine,  eight  hundred  and  seven ;  but  simply  three  hundred  fifty-nine, 
eight  hundred  seven. 

Read  the  following  numbers,  and  state  the  number  of 
hundreds,  tens,  and  ones  in  each  : 

507       469       101       260       206       301       808       888 
321       694       929       300       185       340       671       999 

10.  Notation  and  Numeration.  The  method  of  writing 
numbers  is  called  notation,  and  the  method  of  reading 
numbers  is  called  numeration. 

11.  The  system  of  notation  here  explained  is  called  the 
Arabic  system  of  notation. 

12.  Thousands.  A  group  of  10  hundreds  is  called  a 
thousand,  and  figures  signifying  thousands  are  written  in 
the  fourth  place  from  the  right. 

Thus,  one  thousand,  two  thousands,  three  thousands,  etc.,  are  ex- 
pressed by  1000,  2000,  3000,  etc. 


4  NOTATION   AND   NUMERATION. 

13.  Numbers  expressed  by  Four  Figures.  Numbers 
expressed  by  four  figures  may  be  read  as  thousands, 
hundreds,  tens,  and  ones ;  or  as  hundreds,  tens,  and  ones. 

The  shortest  way  of  reading  numbers  is  the  best  way. 
The  best  way  to  read  1896  is  eighteen  hundred  ninety-six. 
The  best  way  to  read  7005  is  seven  thousand  five. 

Read  in  the  best  way  the  following  numbers : 

1776         1924         1907         2359         5050         3627 
7006         7076         2706         6010         5500         2036 

14.  Orders  of  Units.  The  ones  of  a  number  are  called 
units  of  the  first  order.  The  tens  of  a  number  are  called 
units  of  the  second  order.  The  hundreds  of  a  number  are 
called  units  of  the  third  order.  The  thousands  of  a  number 
are  called  units  of  the  fourth  order. 

Figures  in  the  fifth  place  signify  ten-thousands,  called 
units  of  the  fifth  order.  Figures  in  the  sixth  place  signify 
hundred-thousands,  called  unit*  of  the  sixth  order. 

Note.  The  ones  of  a  number  are  commonly  called  units,  the 
word  units  standing  for  the  phrase  units  of  the  first  order.  Thus, 
we  say  the  number  459  has  4  hundreds,  5  tens,  and  9  units. 

15.  Decimal  System.  Since  ten  units  of  any  order  are 
equal  to  one  unit  of  the  next  higher  order,  this  system 
of  notation  is  called  the  decimal  system;  decimal  being 
derived  from  the  Latin  word  decern,  meaning  ten. 

16.  Periods.  When  the  figures  of  a  number  are  five  or 
more,  we  separate  them  into  groups  of  three  figures  each  by 
commas,  beginning  at  the  right.  The  right-hand  group  is 
called  the  period  of  units ;  the  second  group  is  called  the 
period  of  thousands  ;  the  third  group  is  called  the  period  of 
millions  ;  the  fourth  group  is  called  the  period  of  billions. 

The  unit  of  any  period  is  equal  to  1000  units  of  the  next 
lower  period. 


NOTATION   AND   NUMERATION. 


One  million  is  equal  to  1000  thousands,  and  is  written 
1,000,000;  one  billion  is  equal  to  1000  millions,  and  is 
written  1,000,000,000. 

The  left-hand  period  may  have  one,  two,  or  three  figures  ; 
every  other  period  must  have  three  figures. 

17.   To  Read  an  Integral  Number  expressed  in  Figures. 

Read  the  number  26217320416. 

We  begin  at  the  right  and  point  off  the  figures  into  periods  of 
three  figures  each.     Thus, 

26,217,320,416. 

We  begin  at  the  left  and  read  each  period  as  if  it  stood  alone,  add- 
ing the  name  of  the  period.  The  fourth  period  from  the  right  is  the 
period  of  billions.     Hence  we  read  the  number, 

Twenty-six  billion,  two  hundred  seventeen  million,  three  hundred 
twenty  thousand,  four  hundred  sixteen.     Therefore, 

Beginning  at  the  right,  we  separate  the  figures  by  commas 
into  periods  of  three  figures  each.  We  begin  at  the  left  and 
read  each  period  as  if  it  stood  alone,  adding  the  name  of  each 
period  except  the  name  of  the  period  of  units. 

Note.  The  names  of  periods  above  billions  are  in  order:  trillions, 
quadrillions,  quintillions,  sextillions,  septillions,  octillions,  etc. 

Exercise  1. 


Write  in  periods,  and  read : 


1.  7000. 

2.  7842. 

3.  5043. 

4.  8375. 

5.  2020. 

6.  1753. 

7.  18757. 

8.  75764. 

9.  22003. 
10.  70856. 


11.  234567. 

12.  34561. 

13.  123456. 

14.  654089. 

15.  600897. 

16.  704608. 

17.  350709. 

18.  240682. 

19.  682000. 

20.  753110. 


21.  703101. 

22.  870890. 

23.  21978564. 

24.  17756423. 

25.  300200100. 

26.  707303202. 

27.  3125476890. 

28.  79501346081. 

29.  3000872696. 

30.  72727000000. 


6  NOTATION   AND   NUMERATION. 

18.  To  Write  an  Integral  Number  in  Figures.  Write 
in  figures  the  number  two  hundred  sixty-three  million,  six 
hundred  thirty-five  thousand,  two  hundred  one. 

We  consider  first  the  periods  of  the  number. 

This  number  has  the  period  of  millions,  the  period  of  thousands, 
and  the  period  of  units. 

We  write  first  the  period  of  millions,  and  put  a  comma  after  it ; 
then  the  period  of  thousands,  and  put  a  comma  after  it ;  and  then  the 
period  of  units.     Thus,  263,635,201. 

Note.  Every  period  except  the  one  at  the  left  must  have  three 
figures  (§  16).  If  any  order  of  units  of  a  period  is  lacking,  we  put  a 
cipher  in  its  place  ;  and  if  any  entire  period  is  lacking  we  put  three 
ciphers  in  its  place.  Thus,  four  million,  sixteen  thousand,  four  is 
written  4,016,004  ;  sixteen  million,  four  hundred  sixteen  is  written 
16,000,410.     Therefore, 

We  begin  at  the  left  and  ivrite  the  hundreds,  tens,  and 
units  of  each  period,  putting  zeros  in  all  vacant  places,  and 
putting  a  comma  between  each  period  and  the  period  that 
follows  it. 

Exercise  2. 

Write  in  figures,  arranged  in  periods: 

1.  Six  hundred  thousand,  six. 

2.  Seven   hundred   thirteen   thousand,  three  hundred 
twenty-nine. 

3.  Seven  thousand,  eight  hundred  fifty-four. 

4.  Four  million,  three  thousand,  three  hundred  thirty. 

5.  One  hundred  ten  million,  two  hundred  seventy-nine. 

6.  Nineteen  trillion,  four  million,  three  hundred  nine. 

7.  Seven  million,  six  hundred  seventy-six  thousand, 
four  hundred  sixty-six. 

8.  Three  hundred  forty-seven  million,  six  hundred  fifty- 
one  thousand,  seven  hundred  eighty-five. 

9.  Two  hundred  million,  two  hundred  seven. 

10.    Four  hundred  billion,  four  hundred  thousand,  four. 


NOTATION   AND   NUMERATION.  7 

19.  The  unit  of  measure  of  any  kind  of  quantity  may 
be  divided  into  ten  smaller  measures. 

Thus,  a  dollar,  as  a  measure  of  value,  is  divided  into  ten  dimes, 
or  ten  tenths  of  a  dollar  ;  each  dime  into  ten  cents,  or  ten  hundredths 
of  a  dollar  ;  each  cent  into  ten  mills,  or  ten  thousandths  of  a  dollar. 

20.  The  dollar  sign,  $,  is  written  before  the  number. 

21.  If  dollars  and  cents  are  written,  a  dot  called  the 
decimal  point  is  placed  between  the  dollars  and  cents. 

Thus,  $17  is  17  dollars  ;  $18.20  is  18  dollars  2  dimes,  or  18  dollars 
20  cents  ;  $35,875  is  35  dollars  87  cents  5  mills  ;  and  $0.08  is  8  cents. 

22.  Read  as  dollars,  cents,  and  mills  :  176.375  ;  $1 63.58  ; 
$12.50;  $0,875;  $1.01;  $2.10;  $3.08;  $0.75;  $0,125. 

23.  In  reading  United  States  money,  we  read  the 
number  to  the  left  of  the  decimal  point  as  dollars  ;  the 
number  in  the  first  two  places  to  the  right  of  the  point  as 
cents ;  and  the  number  in  the  third  place  as  mills. 

24.  In  writing  United  States  money,  we  must  have  the 
cents  occupy  two  places.  J^f  the  number  of  cents  is  less 
than  ten,  we  write  a  cipher  in  the  first  place  at  the  right 
of  the  decimal  point. 

25.  Parts  of  other  measures  may  be  expressed  as  tenths, 
hundredths,  etc.;  just  as  dimes,  cents,  and  mills  are 
respectively  tenths,  hundredths,  and  thousandths  of  a 
dollar. 

Thus,  if  we  omit  the  dollar  sign  from  $5,375,  the  expression  may 
stand  for  5  yards,  quarts,  bushels,  or  any  other  full  measures,  and 
375  thousandths  of  another  measure. 

26.  Parts  thus  written  are  called  Decimals.  We  write 
a  number  to  the  right  of  the  units'  place  just  as  we  do  to 
the  left,  first  marking  the  units'  place  with  a  decimal  point 
to  its  right. 


0  NOTATION   AND   NUMERATION. 

27.  In  the  number 

9,876,543,210.123,456,789 

the  full  point  after  0  shows  that  0  stands  in  the  units' 
place.  The  1  to  the  left  is  1  ten,  the  1  to  the  right  is  1 
tenth;  the  2  to  the  left  is  2  hundreds,  the  2  to  the  right  is 
2  hundredtJis ;  the  3  to  the  left  is  3  thousands,  the  3  to  the 
right  is  3  thousandths ;  the  4  to  the  left  is  4  ten-thousands, 
the  4  to  the  right  is  4  ten-thousandtlis ;  the  5  to  the  left  is 
5  hundred-thousands,  the  5  to  the  right  is  5  hundred-thou- 
sandths ;  the  6  to  the  left  is  6  millions,  the  6  to  the  right 
is  6  millionth*  ;  and  so  on. 

Also,  the  210  is  the  units*  period  ;  the  543  is  the  thou- 
sands', the  123  the  thousandths'  period,  etc.;  so  that  the 
number  may  be  read  9  billions,  876  millions,  543  thousands, 
210,  and  123  thousandths,  456  millionths,  789  billionths. 

28.  To  Read  a  Decimal  expressed  in  Figures.  The 
usual  way  of  reading  a  decimal  is  : 

Read  the  decimal  precisely  as  if  a  whole  number,  and 
add  the  fractional  name  of  the  lowest  place. 

Thus,  5.17  is  read  5  and  17  hundredths ;  5.0017,  five  and  17  ten- 
thousandths  ;  6.0203107,  six  and  203  thousand  107  ten-millionths. 

The  word  and  is  distinctly  pronounced  at  the  decimal 
point  and  carefully  omitted  in  all  other  places. 

Thus,  one  hundred  forty-seven  means  147  ;  but  one  hundred  and 
forty-seven  thousandths  means  100.047  ;  and  0.147  must  be  read  one 
hundred  forty-seven  thousandths. 

29.  Practical  computers  often  introduce  the  word 
decimal  at  the  place  of  the  point,  and  then  pronounce  the 
name  of  each  digit  in  succession  to  the  right. 

Thus,  203.07051  may  be  read  two  hundred  three,  decimal,  naught, 
seven,  naught,  five,  one. 


NOTATION   AND   NUMERATION. 


Exercise 

3. 

Read 

1. 

6,728,642. 

16. 

$182,275. 

2. 

3.24658. 

17. 

$0,086. 

3. 

49,568.4782. 

18. 

$0,075. 

4. 

34,598,492,212. 

19. 

$463.87. 

5. 

4,002,000.02. 

20. 

$20,542.02. 

6. 

1872.17. 

21. 

$0.75. 

7. 

94.658265. 

22. 

428,428.428. 

8. 

0.0307. 

23. 

1542.087. 

9. 

100.01. 

24. 

642.873654. 

10. 

1,872,563.372. 

25. 

400.00004. 

11. 

17.008. 

26. 

3,543,362,338. 

12. 

143.00143. 

27. 

0.0000009. 

13. 

29.00081. 

28. 

52.02. 

14. 

5,262,873. 

29. 

56,482.56. 

15. 

8.7854. 

30. 

87,865,842.87866. 

Exercise  4. 
Write  in  figures  :  ' 

1.  Eighty -one  thousand  and  three  hundred  forty-five 
thousandths. 

2.  Thirty-seven   hundred   forty-one   and  six  hundred 
seventy-five  thousandths. 

3.  Four  hundred  thirteen  and  eight  hundredths. 

4.  Ninety-six  and  ninety-six  thousandths. 

5.  Nine  and  forty-three  millionths. 

6.  One   hundred    fifty-four   and    thirty-two   ten-thou- 
sandths. 

7.  Seventy-five  thousandths. 

8.  Three  tenths. 

9.  Forty-four  million,  forty-four  thousand,  forty-four, 
and  forty-four  thousandths. 


10  NOTATION   AND   NUMERATION. 

10.  One  hundred  and  forty-three  millionths. 

11.  One  hundred  forty-three  millionths. 

12.  One  hundred  forty  and  three  millionths. 

13.  Nine  hundred  forty-three  thousand  and  nine  hun- 
dred forty-three  thousandths. 

14.  Seven  hundred  twenty-two  ten-millionths. 

15.  Thirteen,  decimal,  naught,  one,  four,  six,  eight. 

16.  Four  and  one  thousand  nine  ten-thousandths. 

17.  One  hundred  one  and  one   hundred  one  ten-thou- 
sandths. 

18.  Seventeen  million,  six  hundred  forty-nine  thousand. 

19.  Twelve  billion,  twelve  thousand. 

20.  Twelve  billion  and  twelve  thousandths. 

21.  Eight  dollars  and  twelve  cents. 

22.  One  hundred  twenty-seven  dollars  and  one  cent. 

23.  Fourteen  thousand,  two  hundred  seventy-eight  dol- 
lars and  twenty-seven  cents,  five  mills. 

24.  One  thousand  dollars  and  one  cent,  one  mill. 

25.  Two  hundred  thirty-four  dollars  and  fifty-five  cents. 

26.  Twenty-five  cents  ;  three  cents,  four  mills. 

27.  One    million,   four   hundred  eighty-nine  thousand, 
five  hundred  ninety  and  five  hundred  ninety  thousandths. 

28.  Forty-three   thousand,   six   hundred   seventy-seven 
and  four  thousand  six  hundred-thousandths. 

29.  Three  thousand  sixty-nine  and  seventy-eight  thou- 
sand four  hundred  sixteen  ten-millionths. 

30.  The  Roman   System  of   Notation.      The   Roman 
system  employs  seven  letters,  as  follows : 

Letters,        I        V         X         L         C         D         M 
Values,        15         10        50       100      500      1000 

The  first  nine  numbers  are  expressed  by 

I        II        III        IV        V        VI        VII        VIII         IX 
123456  7  8  9 


NOTATION   AND   NUMERATION.  11 

The  tens  are  expressed  by 

X  XX       XXX       XL      L      LX       LXX      LXXX      XC 

10  20  30  40       50      60  70  80  90 

Tens  and  ones  are  expressed  by  writing  the  expressions 
for  units  at  the  right  of  the  expressions  for  tens.     Thus, 

XI  XII       XIV       XV       XIX       XXII       LVI       XCIX 

11  12  14  15  19     •       22  56  99 

The  hundreds  are  expressed  by 

C       CC      CCC      CD       D      DC      DCC      DCCC      DCCCC 
100    200      300      400      500    600       700  800  900 

Writing  M  at  the  left  of  each  of  these  expressions  we 
have 

MC   MCC   MCCC  MCD  MD  MDC  MDCC  MDCCC  MDCCCC 

1100   1200      1300     1400   1500    1600      1700         1800  1900 

Hundreds,  tens,  and  ones  are  expressed  by  writing  the 
hundreds,  then  the  tens,  and  then  the  ones. 

Thus,  eighteen  hundred  ninety-six  is  written:  MDCCC  for  eighteen 
hundred,  then  XC  for  ninety,  and  VI  for  six,  making  MDCCCXCVI. 

Exercise  5. 
Read  : 

XXXVI;  XL;  XLVI ;  LVIII ;  LIX  ;  LXXXI;  XCI ; 
XCIII ;  CIX  ;  CCIX  ;  CCXX  ;  CLIX  ;  MDCCCLXXXVI ; 
MDCLXVI ;  MDCCLXXVI ;  MCDLIX  ;  MDLXXXIX. 

Express  in  the  Roman  system  : 

43;  55;  81;  77;  99;  113;  128;  514;  724 ;  630 ;  1020 ; 
1040 ;  1088  ;  1131  ;  1218  ;  1492  ;  177(5  ;  1899  ;  1319  ;  1556 ; 
1897;  1620;  1783;  1812;  1861 ;  1872. 


CHAPTER   II. 

ADDITION  AND  SUBTRACTION. 

31.  The  sign  +  is  called  plies,  and  means  that  the 
number  after  it  is  to  be  counted  with  the  number  before  it ; 
that  is,  added  to  the  number  before  it. 

32.  The  sign  =  is  called  the  sign  of  equality,  and  stands 
for  the  word  equals ;  so  that  5  +  4  =  9  is  read  :  5  plus  4 
equals  9. 

33.  Addition.  The  operation  of  finding  a  number  equal 
to  two  or  more  numbers  taken  together  is  called  addition  .- 
and  the  result  is  called  their  sum. 

34.  The  sum  of  two  or  more  numbers  is  the  same  in 
whatever  order  the  numbers  are  added. 

Thus,  3  +  2  +  5  =  10,     or    5  +  3  +  2  =  10. 

35.  Abstract  Numbers.  Numbers  not  applied  to  any 
particular  unit,  as  7,  17,  25,  are  called  abstract  numbers. 

36.  Concrete  Numbers.  Numbers  applied  to  a  particular 
unit,  as  7  horses,  17  apples,  are  called  concrete  numbers. 

37.  Like  Numbers.  Numbers  applied  to  the  same  unit. 
expressed  or  understood,  are  called  like  numbers. 

38.  Only  like  numbers,  and  units  of  the  same  order,  can 
be  added. 

Exercise  6. 

Count  to  100  or  more : 

1.  By  2's,  beginning  0,  2,  4 ;  beginning  1,  3,  5. 

2.  By  3's,  beginning  0,  3,  6  ;  beginning  1,  4,  7  ;  beginning 
2,  5,  8. 


ADDITION    AND   SUBTRACTION. 


13 


3.  By  4's,  beginning  0,  4,  8 ;  beginning  1,  5,  9 ;  begin- 
ning 2,  6,  10 ;  beginning  3,  7,  11. 

4.  By  5's,  beginning  0,  5,  10  ;  beginning  1,  6, 11 ;  begin- 
ning 2,  7,  12 ;  beginning  3,  8,  13 ;  beginning  4,  9,  14. 

5.  By  6's,  beginning  0,  6,  12  ;  beginning  1,  7,  13  ;  begin- 
ning 2,  8,  14 ;  beginning  3,  9,  15 ;  beginning  4,  10,  16  ; 
beginning  5,  11,  17. 

6.  By  7's,  beginning  0,  7,  14 ;  beginning  1,  8,  15  ;  begin- 
ning 2,  9,  16  ;  beginning  3,  10,  17 ;  beginning  4,  11,  18 ; 
beginning  5,  12,  19;  beginning  6,  13,  20. 

7.  By  8's,  beginning  0,  8,  16 ;  beginning  1,  9,  17  ;  begin- 
ning 2,  10,  18;  beginning  3,  11,  19;  beginning  4,  12,  20; 
beginning  5,  13,  21  ;  beginning  6,  14,  22  ;  beginning  7, 
15,23. 

8.  By  9's,  beginning  0,  9, 18  ;  beginning  1, 10, 19  ;  begin- 
ning 2,  11,  20  ;  beginning  3,  12,  21 ;  beginning  4,  13,  22 ; 
beginning  5,  14,  23  ;  beginning  6,  15,  24 ;  beginning  7,  16, 
25  ;  beginning  8,  17,  26. 


Find  the  sum  of 

9.   10. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

3      2 

3 

5 

3 

2 

5 

5 

4 

5 

3 

1 

5      1 

6 

6 

3 

7 

3 

6 

8 

5 

.    6 

8 

7      9 

7 

7 

4 

7 

2 

4 

7 

3 

7 

8 

6      8 

8 

8 

5 

3 

1 

7 

3 

6 

3 

7 

21.  22. 

23. 

24. 

25. 

26. 

27. 

28. 

29. 

30. 

81. 

32. 

6      9 

6 

4 

4 

3 

6 

7 

5 

8 

2 

9 

8      5 

4 

5 

4 

7 

2 

5 

5 

2 

9 

6 

7      4 

3 

6 

3 

5 

1 

8 

9 

2 

9 

5 

9      3 

7 

7 

7 

5 

8 

3 

3 

7 

4 

4 

Note.  In  adding,  begin  at  the  bottom,  and  give  results  only. 
Thus,  in  example  9,  say  13  18,  21;  do  not  say  6  and  7  are  13,  13  and 
5  are  18,  and  18  and  3  are  21. 


14  ADDITION    AND   SUBTRACTION. 

39.  Examples.  1.  Find  the  sum  of  45.32,  27.21,  and 
68.55. 

Solution.  Since  only  units  of  the  same  order  can  be  added,  we 
write  units  under  units,  tens  under  tens,  tenths  under  tenths,  and 
hundredths  under  hundredths,  so  that  the  decimal  points  stand  in  a 
vertical  line  ;  and  draw  a  line  beneath. 

We  first  find  the  sum  of  the  hundredths  and  write  this  sum,  8, 

under  the  column  of  hundredths.     We  next  find  the  sum  of 

45.32    the  tenths  and  write  the  units'  figure,  0,  of  this  sum  under 

27.21     the  column  of  tenths,  but  the  tens'  figure,  1,  of  this  sum  we 

68.55    add  with  the  figures  of  the  next  column,  making  1  +  8  +  7  + 

141.08    5  =  21.     We  write  the  units'  figure,  1,  of  this  sum  under 

the  column  of  units,  but  the  tens'  figure,  2,  we  add  with  the 

figures  of  the  next  column,  making  2+6  +  2  +  4  =  14.     We  write 

the  entire  sum  of  the  last  column  on  the  left.     We  put  the  decimal 

point  in  the  sum  directly  under  the  column  of  decimal  points. 

2.    Add  41.38,  37.89,  26.67,  and  58.21. 

41.38  In  adding,  the  following  wording,  and  no  more,  should  be 

37.89  used  :  one,  eight,  seventeen,  twenty-five  (emphasize  five,  and 

26.67  write  it  while  pronouncing  it),  carry  two  ;  four,  ten,  eigh- 

58.21  teen,  twenty-one,  carry   two ;  ten,   sixteen,   twenty -three, 

164.15  twenty-four,  carry  two  ;  seven,  nine,  twelve,  sixteen. 

40.  Hence,  we  have  the  following 

Rule  for  Addition.  Write  the  numbers  so  that  units 
of  the  same  order  shall  stand  in  the  same  column. 

Add  the  right-hand  column;  write  the  units  of  this  sum 
beneath,  and  add  the  tens,  if  any,  to  the  next  column. 

So  proceed  with  each  column.  Write  the  entire  sum  of  the 
last  column. 

If  any  of  the  numbers  contain  a  decimal,  write  the  decimal 
point  in  the  sum  directly  under  the  column  of  decimal  points 
in  the  numbers. 

Proof.  Add  each  column  in  reverse  order.  If  the  results 
agree,  the  work  may  be  assumed  to  be  correct. 


ADDITION   AND    SUBTRACTION. 


15 


Exercise  7. 


Find  the  sum  of  : 

1. 

231  +  764. 

8. 

2. 

341  +  57.8. 

9. 

3. 

430.31  +  58.61. 

10. 

4. 

512.87  +  36.84. 

11. 

5. 

12.78+711.56+415.86. 

12. 

6. 

1543.1+164.7. 

13. 

7. 

1728  +  402.56. 

14. 

1897.3+675.34+6897.65. 
475.34  +  6897.65  +  1728. 
402.56  +  164.7  +  0.5236. 
0.7854+3.1416+2.71828. 
2.71828+402.56  +  1897.3. 
0.7854  +  4.12  +  30.103. 
2.7113  +  27.53  +  341.586. 


15.  230.8  +  223  +  2.63  +  373.8. 

16.  32.358  +  821.9  +  23.04+73.7. 

17.  202.3031  +  71.575  +  65.813. 

18.  0.0078  +  7.377+653.03. 

19.  653.03  +  65.303  +  6.5033. 

20.  939.303  +  65.746  +  8.2794. 


21. 

22. 

23. 

2.7182818 

0.4342945 

1.6093295 

3.1415927 

0.2098882 

15.4323487 

0.7853982 

4.81P4774 

3.785 

24. 

25. 

26. 

0.4771213 

1.6093295 

0.6213768 

0.2908882 

3.2808693 

3.785 

4.8104774 

0.3937043 

0.264 

2.5399772 

0.5235988 

15.4323487 

0.3937043 

0.4342945 

1.7320508 

27. 

28. 

29. 

0.6213768 

0.3937043 

1.4142136 

1.4142136 

0.3047973 

1.6093295 

3.2808693 

1.7320508 

0.30103 

0.3047973 

2.236068 

0.381966 

4.8104774 

0.381966 

3.2808693 

16 


ADDITION    AND    SUBTRACTION. 


41.  A  simple  test  of  the  correctness  of  an  addition  is  to 
add  a  second  time,  beginning  at  the  top  instead  of  the  bot- 
tom of  the  columns,  or  to  add  two  columns  at  once. 

It  is  of  great  advantage  to  educate  the  eye  to  take  in  at 
a  glance  digits  enough  to  make  10  or  more,  and  then  these 
sums  can  be  added  instead  of  the  separate  digits. 

To  illustrate,  take  the  example  in  the  margin.  Adding  from  the 
bottom,  the  computer  says  to  himself  (seeing  8  +  2  =  10, 

61,803    and  9+  3  =  12),  10,  12,  22;  8  ;  8,  12,  20  ;  11,  4,  15  ;  11, 

43,429     10,  21. 

47,712        In    the    two-column   mode  he  says,   50,   32,   82;  and 

62,138  writes  down  the  82.  Then  he  continues,  98,  52,  150,  and 
215,082  writes  the  50  at  the  left  of  82  ;  then  he  proceeds,  11, 
10,  21,  and  writes  the  21. 

Many  bookkeepers  and  merchants  strongly  recommend 
the  addition  of  two  columns  at  once,  as  the  most  expedi- 
tious and  the  least  liable  to  error. 

42.  Many  computers  begin  at  the  bottom  of  the  right- 
hand  column  in  adding,  and  write  on  a  piece  of  waste-paper 
the  full  sum  of  each  column  or  double  column ;  then  they 
begin  at  the  top  of  the  left-hand  column,  and  add  each  col- 
umn or  double  column,  also  writing  the  full  sum  ;  finally, 
they  add  the  sums  obtained  in  the  first  addition,  and  the 
sums  obtained  in  the  second  addition,  and  compare  the 
results. 


Thus,  in  the  example  above, 
By  Single  Columns : 


22 

20 

6 

13 

20 

20 

13 

6 

20 

22 

215082 

215082 

By  Double  Columns 


82 


150 


215082 


213 
206 
22 

215082 


ADDITION    AND    SUBTRACTION.  17 

Exercise  8. 

Add  the  following  by  double  columns,  and  test  by  adding 
by  single  columns : 

1.  2.  3. 

$  45.68  $154.31  $  73.86 

73.91  296.85  453.71 

78.54  736.48  137.64 

534.69  345.19  98.87 

134.70  782.34  643.48 
581.43  78.43  462.71 


6. 

$621.65 
167.32 
856.96 
718.83 
501.49 
315.72 
768.44 

9. 

$763.89 

78.23 

345.61 

26.73 

489.56 

812.35 

607.28 

219.07 

68.72 

216.78 

436.74 


4. 

5. 

$498.50 

$  65.42 

17.37 

638.34 

684.29 

763.43 

231.56 

809.31 

210.10 

798.83 

671.54 

835.78 

643.53 

356.47 

7. 

8. 

$791.52 

$  32.54 

504.83 

254.63 

879.26 

63.27 

243.97 

131.56 

732.86 

506.72 

47.95 

283.54 

856.43 

345.83 

497.65 

643.46 

541.26 

708.91 

616.72 

463.73 

857.94 

67.74 

18 


ADDITION    AND    SUBTRACTION. 


10. 


11 


12. 


$8400.07 

$1873.33 

$2336.29 

3212.17 

6170.24 

336.00 

1716.41 

4813.25 

2456.25 

1020.08 

662.25 

641.25 

1452.44 

622.64 

1174.50 

1829.51 

692.82 

326.03 

1929.96 

2457.75 

1219.87 

114.78 

2126.76 

226.78 

89.75 

5391.25 

276.75 

173.67 

7349.86 

5936.40 

17.45 

l  122.75 

1914.78 

112.44 

9667.50 

311.87 

1098.75 

6000.00 

7956.00 

6170.24 

572.80 

1919.66 

13. 


14. 


15. 


$1482.40 

$  773.72 

$2406.08 

2575.71 

442.37 

3101.24 

3364.27 

454.86 

1452.09 

689.81 

358.61 

3693.91 

1533.61 

2003.17 

2054.76 

735.58 

179.56 

1231.25 

105.69 

8493.75 

1828.35 

261.64 

4179.54 

1562.50 

1516.56 

3493.54 

6937.50 

2197.23 

178.17 

1987.57 

1317.71 

727.53 

943.27 

408.30 

2889.42 

2312.11 

609.53 

992.92 

1409.28 

1679.47 

1183.08 

2759.94 

ADDITION    AND   SUBTRACTION.  19 

43.  The  sign  —  is  called  minus,  and  means  that  the 
number  after  it  is  to  be  taken  from  the  number  before  it ; 
that  is,  subtracted  from  the  number  before  it. 

The  expression  9  —  5  =  4  is  read  :    9  minus  5  equals  4. 

44.  Subtraction.  The  process  of  taking  one  number 
from  another  is  called  subtraction. 

45.  The  number  taken  away  is  called  the  subtrahend; 
the  number  from  which  the  subtrahend  is  taken,  the  minu- 
end; and  the  number  remaining,  the  remainder  or  difference. 

46.  The  sum  of  the  remainder  and  the  subtrahend  is 
always  equal  to  the  minuend.     Hence, 

47.  To  test  the  accuracy  of  the  work  in  subtraction,  we 
add  the  remainder  and  the  subtrahend.  The  sum  will  be 
equal  to  the  minuend,  if  the  work  is  correct. 

48.  The  minuend,  subtrahend,  and  remainder  must  all  be 
like  numbers  ;  and  from  units  of  any  order  units  of  the  same 
order  only  can  be  subtracted,  ones  from  ones,  tens  from  tens, 
tenths  from  tenths,  hundredths  from  hundredths,  etc. 

Exercise  9. 

1.  Subtract  by  2's  from  20  to  0 ;  from  21  to  1. 

2.  Subtract  by  3's  from  20  to  2  ;  from  21  to  0. 

3.  Subtract  by  4's  from  30  to  2  ;  from  31  to  3  ;  from  32 
to  0  ;  from  33  to  1. 

4.  Subtract  by  5's  from  32  to  2 ;  from  33  to  3  ;  from  34 
to  4  ;  from  35  to  0 ;  from  36  to  1. 

5.  Subtract  by  6's  from  33  to  3  ;  from  34  to  4 ;  from  35 
to  5  ;  from  36  to  0  ;  from  37  to  1  ;  from  38  to  2. 

6.  Subtract  by  7's  from  42  to  0  ;  from  43  to  1  ;  from  44 
to  2  ;  from  45  to  3  ;  from  46  to  4  ;  from  47  to  5. 

7.  Subtract  by  8's  from  42  to  2  ;  from  43  to  3  ;  from  44 
to  4  ;   from  45  to  5  ;  from  46  to  6  ;   from  47  to  7. 

8.  Subtract  by  9's  from  55  to  1 ;  from  56  to  2  ;  from  57 
to  3 ;  from  59  to  5 ;   from  61  to  7  ;   from  62  to  8. 


20  ADDITION    AND    SUBTRACTION. 

49.  Examples.     1.  From  359.7  take  186.3. 

Solution.    We  write  units  under  units,  tens  under  tens,  tenths  under 

tenths,  and  so  on.    Then  3  tenths  from  7  tenths  leaves  4  tenths,  and  we 

write  4  under  the  column  of  tenths ;  6  units 

Minuend,         359.7    from  9  units  leaves  3  units,  and  we  write  3 

Subtrahend,     186.3    under  the  column  of  units ;  since  we  cannot 

Remainder,      173.4    take  8  tens  from  5  tens,  we  change  one  of  the 

3  hundreds  to  10  tens  and  add  them  to  the  5 

tens,  making  15  tens;  then  8  tens  from  15  tens  leaves  7  tens,  and  we 

write  7  under  the  column  of  tens.     As  we  have  taken  one  of  the  3 

hundreds,  we  have  only  2  hundreds  remaining ;  and  1  hundred  from 

2  hundreds  leaves  1  hundred.     The  remainder,  therefore,  is  173.4. 

2.    From  50  take  27.65. 

Solution.     Since  the  subtrahend  contains  tenths  and  hundredths, 

and  the  minuend  has  neither  tenths  nor  hundredths,  we  put  zeros  in 
the  place  of  tenths  and  hundredths  in  the  minuend.  As 
(4)(9)(9)(10)  there  are  no  hundredths,  no  tenths,  and  no  units  in  the 
5  0.  0  0  minuend,  1  of  the  6  tens  is  taken,  leaving  4  tens,  and 
2  7.  6  5  changed  to  10  units  ;  then  1  of  the  10  units  is  taken, 
2  2.  3  5  leaving  9  units,  and  changed  to  10  tenths  ;  then  one  of  the 
10  tenths  is  taken,  leaving  9  tenths,  and  changed  to  10 

hundredths.    That  is,  50.00  is  changed  to  4  tens,  9  units,  9  tenths,  and 

10  hundredths.     Then,  subtracting,  we  have  22.35. 

50.  Hence,  we  have  the  following 

Rule  for  Subtraction.  Write  the  subtrahend  under  the 
minuend,  placing  units  of  the  sams  order  in  the  same  column. 

Begin  at  the  right  and  subtract  each  order  of  units  of  the 
subtrahend  from  the  corresponding  order  of  the  minuend. 
Write  the  result  beneath,  step  by  step,  and  put  in  the  decimal 
point  when  reached. 

If  any  order  of  the  minuend  has  fewer  units  than  the 
same  order  of  the  subtrahend,  increase  the  units  of  this  order 
of  the  minuend  by  10  and  subtract ;  then  diminish  by  one 
the  units  of  the  next  higher  order  of  the  minuend. 

Proof.  Add  the  remainder  and  subtrahend.  If  the  sum 
equals  the  minuend,  the  work  may  be  assumed  correct. 


ADDITION   AND   SUBTRACTION. 


21 


Exercise  10. 


Find  the  remainder  and  prove  : 

1.  234-123.  11.   789-456. 

2.  343-123.  12.   879-456. 

3.  424-123.  13.   978-456. 

4.  555-123.  14.    6378-456. 

5.  676-123.  15.    6855-456. 

6.  725-123.  16.   6853-456. 

7.  839  —  123.  17.    7797  —  456. 

8.  999-123.  18.    7006-456. 

9.  1000  —  123.  19.   3542  —  456. 
10.  5120  —  123.  20.    4000  —  456. 

31.  $183.45 -$76.47.  51. 

32.  $716.43  —  $628.74.  52. 

33.  $647.51 -$549.64.  53. 

34.  $270.04— $128.31.  54. 

35.  $125  — $101.50.  55. 

36.  $247.93  — $129.47.  56. 

37.  $641.87  -  $333.95.        '  57. 

38.  $56.27  — $29.89.  58. 

39.  3.1415927-2.7182818.  59. 

40.  0.7853982  —  0.5235988.  60. 

41.  4.8104774  —  0.4342945.  61. 

42.  2.5399772  —  0.3937043.  62. 

43.  0.3937043-0.3047973.  63. 

44.  3.2808693  —  0.3047973.  64. 

45.  3.2808693-1.6093295.  65. 

46.  3.785—0.6213768.  66. 

47.  15.4323487  —  0.264.  67. 

48.  1.7320508  —  1.4142136.  68. 

49.  2.236068-1.7320508.  69. 

50.  2.236068  —  0.618034.  70. 


21.  974-779. 

22.  368  —  249. 

23.  2301  —  479. 

24.  2731  —  929. 

25.  708-394. 

26.  1123  —  1072. 

27.  891  —  773. 

28.  8103  —  5621. 

29.  19.001  —  3456. 

30.  2180-792. 
0.381966  -  0.30103. 
3.1415927-0.7853982. 
2.3561945  -  0.7853982. 
1.5707963-0.7853982. 
3.1415927  —  0.5235988. 
2.6179939-0.5235988. 
2.0943951  —  0.5235988. 
1.5707963  -  0.5235988. 
1.0471975-0.5235988. 
1-0.381966. 
1.4142136-0.618034. 
0.618034-0.381966. 
9,873,210  —  8,765,420. 
8010.101  -  4187.94. 
1,000,000  -  817,259. 
729,434-613,488. 
6532.18  —  1916.47. 
1718.754  —  1389.328. 
21,205  —  1787.563. 
42,786.95-4278.695. 


22  ADDITION    AND    SUBTRACTION. 

Exercise  11. 

1.  In  a  till  are  $391  in  bills,  $67.50  in  gold,  $39.75  in 
silver,  and  $2.77  in  copper  and  nickel.  How  much  money 
is  in  the  till  ? 

2.  Starting  out  with  $315.75  in  one  wallet  and  $54.37 
in  another,  I  pay  the  grocer  $127.38  ;  the  butcher,  $64.17 ; 
the  shoemaker,  $21.40 ;  the  landlord,  $50  ;  the  tailor,  $35. 
What  ought  I  to  have  left? 

3.  On  a  bill  of  $753.43,  I  pay  $517.87.  How  much  do 
I  still  owe?  If  I  owe  $817.87,  and  have  but  $637.50,  how 
much  do  I  lack  of  being  able  to  pay  ? 

4.  If  a  man  was  born  January  1, 1812,  how  old  was  he 
January  1,  1878? 

5.  America  was  discovered  in  1492.  How  many  years 
after  its  discovery  was  each  of  the  following  events  ? 

Settlement  of  Florida,  1565 ;  of  Virginia,  1607  ;  of  Mas- 
sachusetts, 1620 ;  of  Quebec,  1608 ;  French  and  Indian 
War,  1756 ;  Declaration  of  Independence,  1776  ;  Inaugura- 
tion of  Washington,  1789  ;  War  with  England,  1812  ;  Mexi- 
can War,  1846 ;  Civil  War,  1861. 

6.  The  minuend  is  one  hundred  million  two  hundred 
fifty-six  thousand  three  hundred  seventy-two,  and  the  sub- 
trahend is  nineteen  million  nine  hundred  thousand  nine 
hundred  ninety-nine.     Find  the  remainder. 

7.  If  the  minuend  is  9874,  and  remainder  3185,  what  is 
the  subtrahend  ?  The  subtrahend  being  7659,  and  re- 
mainder 675.68,  what  is  the  minuend  ? 

8.  The  smaller  of  two  numbers  is  7.9;">764328  ;  their 
difference  is  0.00087692.     What  is  the  larger  number  ? 

9.  The  larger  of  two  numbers  is  7.95764328,  and  their 
difference  is  7.153485.     What  is  the  smaller  number? 

10.  If  the  subtrahend  is  10,542,  and  the  difference  544.2, 
what  is  the  minuend  ? 


ADDITION   AND    SUBTRACTION.  23 

11.  A  man  pumps  out  of  a  cistern  in  one  hour  243.75 
gallons  ;  in  the  next  hour,  227.5  gallons  ;  in  45  minutes 
more,  137.75  gallons  ;  and  the  cistern  is  empty.  How- 
many  gallons  of  water  were  in  it? 

12.  From  what  number  must  I  subtract  5  to  leave  7? 
8  to  leave  9  ?  From  what  number  must  I  subtract  5.1736 
to  leave  8.1964?  6.231  to  leave  9.6648?  74.213  to  leave 
25.787  ? 

13.  What  must  be  subtracted  from  1  to  leave  0.5  ?  to  leave 
0.53  ?  to  leave  0.532  ?  to  leave  0.5236  ?  to  leave  0.5235988  ? 

14.  I  start  on  a  journey  of  3433  miles.  The  first  day 
I  make  428  miles  ;  the  second  day,  511  miles ;  the  third, 
497  miles  ;  the  fourth,  513.  How  many  miles  of  my  jour- 
ney remained  for  me  at  the  close  of  each  day  ?  How  many 
miles  had  I  gone  at  the  close  of  each  day  ? 

15.  Subtract  76,343  from  the  sum  of  61,932,  51,387, 
5193,  4674,  and  8199  ;  then  subtract  23,657  from  the  re- 
mainder. 

16.  Jones  bought  a  farm  and  stock  for  $7633.90  ;  sold 
the  stock  for  $305.75;  then  sold  the  farm  for  $7325. 
How  much  did  he  lose  ?        ' 

17.  If  I  gave  $4375  for  my  land,  and  paid  for  house, 
barn,  sheds,  and  fences,  $2789.50  ;  also  $973.75  for  horses, 
cattle,  tools,  etc. ;  what  did  my  farm  and  stock  cost  ? 

18.  If  I  paid  $8138.25  for  land  and  cattle,  and  sold 
part  of  the  land  for  $675,  and  part  of  the  cattle  for  $217.50, 
what  is  the  cost  of  the  land  and  the  cattle  left  ? 

19.  John  has  158  cents,  James  has  271  cents;  James 
gives  John  56  cents.  Which  has  then  more  than  the 
other,  and  how  many  cents  more  ? 

20.  A  cattle  dealer  had  228  oxen,  475  sheep,  and  49 
lambs;  he  sold  17  oxen,  64  sheep,  and  7  lambs.  How 
many  animals  of  each  kind  did  he  then  have,  and  how 
many  all  together  ? 


CHAPTER   III. 
MULTIPLICATION. 

51.  Multiplication.  The  process  of  taking  a  number  of 
units  a  number  of  times  is  called  multiplication. 

52.  Multiplicand.  The  number  of  units  taken  is  called 
the  multiplicand. 

53.  Multiplier.  The  number  that  shows  how  many 
times  the  multiplicand  is  taken  is  called  the  multiplier. 

54.  Product.  The  number  found  by  multiplication  is 
called  the  product. 

55.  The  multiplier  always  signifies  a  number  of  times, 
and  is,  therefore,  an  abstract  number. 

56.  The  multiplicand  and  product  are  like  numbers. 

57.  Factors.  The  numbers  used  in  making  a  product 
are  called  factors  of  the  product. 

58.  The  product  of  two  factors  is  the  same  whichever 
factor  is  taken  as  the  multiplier. 

•  •    •    •       Thus,  3  times  4  =  4  times  3.     The  dots  in  the  lnar- 

•  •   •   •   gin  read  across  the  page  make  3  fours ;  read  up  and 

•  •   •   •    down  the  page  they  make  4  threes. 

Note.  The  multiplicand  always  signifies  a  number  of  units, 
whether  the  kind  of  units  is  stated  or  not.  The  only  difference 
between  15  and  15  horses  is  that  in  the  first  case  the  kind  of  units 
counted  is  not  stated,  and  in  the  second  case  the  kind  is  stated. 

We  may  interchange  the  multiplicand  and  multiplier  if  we  refer  to 
the  numbers  only.  Thus,  in  the  example  3  times  4  horses,  we  cannot 
say  4  horses  times  3,  but  we  may  interchange  the  3  and  4,  and  have 
4  times  3  horses.  The  product  in  either  case  is  12  horses.  With  this 
understanding,  we  may  always  use  the  smaller  number  as  multiplier. 


MULTIPLICATION. 


25 


59.  The  sign  of  multiplication  is  X.  When  the  multi- 
plier precedes  the  multiplicand,  the  sign  X  is  read  times. 

Thus,  6  X  $7  =  $42  is  read  :  6  times  $7  equals  $42. 

When  the  multiplier  follows  the  multiplicand,  the  sign  X 
is  read  multiplied  by. 

Thus,  $7X6=  $42  is  read :  $7  multiplied  by  6  equals  $42,  and 
means  $7  taken  6  times  equals  $42. 

60.  The  products,  in  all  cases  in  which  neither  factor 
exceeds  twelve,  should  be  thoroughly  committed  to  memory. 
They  will  be  found  in  the  following 

Multiplication  Table. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20. 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 
132 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

12 

24 

36 

48 

60 

12 

84 

96 

108 

120 

132 

144 

61.  In  this  table,  take  the  multiplicand  in  the  upper 
line,  and  the  multiplier  in  the  left-hand  column;  the 
product  will  be  found  directly  under  the  multiplicand, 
and  opposite  the  multiplier  ;  as,  12  X  7  is  84. 


26  MULTIPLICATION. 

62.  A  change  in  the  position  of  the  decimal  point  of  a 
number  expressed  in  figures  affects  the  value  of  the 
number. 

Thus,  if  in  79.253  we  move  the  decimal  point  one  place  to  the 
right,  so  that  the  number  becomes  792.53,  we  increase  the  value  of 
each  figure  tenfold ;  the  7  tens  become  7  hundreds,  the  9  units  be- 
come 9  tens,  the  2  tenths  become  2  units,  the  6  hundredths  become  6 
tenths,  and  the  3  thousandths  become  three  hundredths.  The  value 
of  the  entire  number,  therefore,  is  increased  tenfold.  But  multiply- 
ing a  number  by  10  increases  its  value  tenfold.  Hence,  moving  the 
decimal  point  of  a  number  one  place  to  the  right  has  the  same  effect 
as  multiplying  the  number  by  10.  In  the  same  way,  moving  the 
decimal  point  two  places  to  the  right  multiplies  the  number  by 
100,  and  so  on.     Hence, 

63.  To  Multiply  a  Number  by  io,   ioo,   iooo,   etc., 
Move  the  decimal  point  in  the  multiplicand  as  many  ntmeei 

to  the  right,  annexing  zeros  if  necessary,  as  there  are  zeros  in 
the  multiplier. 

Again,  if  in  79.253  we  move  the  decimal  point  one  place  to  the 
left,  so  that  the  number  becomes  7.9253,  we  decrease  the  value  of 
each  figure  tenfold ;  the  7  tens  become  7  units,  the  9  units  become  9 
tenths,  the  2  tenths  become  2  hundredths,  the  5  hundredths  become  5 
thousandths,  and  the  3  thousandths  become  3  ten-thousandths.  The 
value  of  the  resulting  number,  therefore,  is  one  tenth  that  of  the 
original  number.  But  multiplying  a  number  by  0.1  means  to  find 
one  tenth  of  the  number.  Hence,  moving  the  decimal  point  of  a 
number  one  place  to  the  left  has  the  same  effect  as  multiplying  the 
number  by  0.1.  In  the  same  way,  moving  the  decimal  point  two 
places  to  the  left  has  the  same  effect  as  multiplying  the  number  by 
0.01,  and  so  on.     Hence, 

64.  To  Multiply  a  Number  by  o.i,  o.oi,  o.ooi,  etc., 

More  the  decimal  point  in  the  multiplicand  as  many  places 
to  the  left,  prefixing  zeros  if  necessary,  as  there  are  decimal 
places  in  the  multiplier. 


MtTLTtFLICATtON.  27 

65.  Since  multiplication,  as  defined  in  §  51,  is  the  process 
of  taking  the  multiplicand  the  number  of  times  indicated 
by  the  multiplier,  the  multiplier  must  be  an  integral  num- 
ber ;  but  the  meaning  of  multiplication  is  extended  to  cover 
the  case  in  which  the  multiplier  contains  a  decimal. 

To  multiply  by  a  decimal  is  to  take  such  a  part  of  the 
multiplicand  as  the  decimal  is  of  one. 

To  multiply  by  an  integral  number  and  a  decimal  is  to 
take  the  multiplicand  as  many  times  as  is  indicated  by  the 
integral  number,  and  such  a  part  of  the  multiplicand  as  is 
indicated  by  the  decimal. 

66.  Examples.     1 .  Find  the  product  of  6  X  34.87. 

Solution.     We  write  the  multiplier,  6,  under  the  multiplicand,  as 
in  the  margin,  and  begin  at  the  right  to  multiply.    6  times  7  hun- 
dredths equals  42  hundredths,  or  4  tenths  and  2  hun- 
34.87    dredths.     We  write  the  2  hundredths  in  the  place   of 

6    hundredths,  and  reserve  the  4  tenths  to  add  to  the  product 

209.22  of  the  tenths.  6  times  8  tenths  equals  48  tenths,  and  48 
tenths  plus  the  4  tenths  reserved  makes  52  tenths,  or 
5  units  and  2  tenths.  We  write^  the  2  tenths  in  the  place  of  tenths 
and  reserve  the  5  units  to  add  to  the  product  of  the  units.  6  times 
4  units  equals  24  units,  and  24  units  plus  the  5  units  reserved  makes 
29  units,  or  2  tens  and  9  units.  We  write  the  9  units  in  the  place  of 
units,  and  reserve  the  2  tens  to  add  to  the  product  of  the  tens.  6 
times  3  tens  equals  18  tens,  and  18  tens  plus  the  2  tens  reserved  makes 
20  tens.     We  write  20  to  the  left  of  the  9  units. 

2.    Find  the  product  of  0.6  X  34.87. 

Solution.      The  multiplier,  0.6,  equals  6  X  0.1.     We 

34.87     therefore  multiply  first  by  6,  and  this  product  by  0.1. 

0.6    But  multiplying  by  0.1,  simply  moves  the  decimal  point 


20.922     in  the  product  one  place  to  the  left.     Hence,  the  product 
has  two  decimal  places  for  the  decimal  in  the  multiplicand, 
and   one  more  place   for  the    decimal  in   the  multiplier,   or   three 
in  all. 


28 


MULTIPLICATION. 


3.    Find  the  product  of  263  X  74.782. 

Solution.  The  multiplier  is  200  +  60  +  3.  We  obtain  the  product 
by  multiplying  the  multiplicand  by  3,  then  by  60,  then  by  200,  and 
adding  these  products. 


(1) 

74.782 
263 


(2) 

74.7*2 


224.346 

224346 

4486.920 

448692 

14956.400 

149664 

19067.666        19667.666 


3  times  the  multiplicand  = 
60  times  the  multiplicand  = 
200  times  the  multiplicand  = 
263  times  the  multiplicand  = 
Since  zeros  at  the  right  of  the  partial  products  do  not  affect  the 
result  of  the  addition,  they  may  be  omitted  as  in  (2).     Care  must  be 
taken,  however,  to  put  the  right-hand  figure  of  each  partial  product 
directly  under  the  figure  of  the  multiplier  used  in  obtaining  it. 

4.  Find  the  product  of  2.63  X  74.782. 

74.782  Solution.     The  multiplier,  2.63,  equals  263  X  0.01. 

2.63  We  therefore  multiply  first  by  263  and  this  product  by 

224346  0.01.     But   multiplying  by  0.01  simply  moves  the  deci- 

448(592  mal  point  in  the  product  two  places  to  the  left.    Hence, 

149564  the  product  has  three  decimal  places  for  the  decimal 

19(5.67666  in  the  multiplicand,  and  two  more  places  for  the  deci- 
mal in  the  multiplier,  or  five  in  all. 

5.  Find  the  product  of  2007  X  4587. 

4587 

2007        Solution.    The  partial  products  corresponding  to  the 
32109    zeros  in  the  multiplier  will  be  zero,  and  therefore  they 
91 7  \  need  not  be  written. 

U2W5109 

67.  In  each  of  these  examples,  if  we  interchange  the 
multiplier  and  multiplicand  and  multiply,  we  find  that  we 
obtain  the  same  product. 

68.  Hence,  we  have  the  following 

Rule  for  Multiplication.  Write  the  multiplier  under 
the  multiplicand  with  their  right-hand  figures  in  a  vertical 
line,  and  draw  a  line  beneath. 


MULTIPLICATION. 


29 


Begin  at  the  right  and  multiply  each  order  of  units  of  the 
multiplicand  by  each  order  of  the  multiplier.  Write  the  units 
of  each  product,  and  add  the  tens,  if  any,  to  the  next  product. 

Place  the  right-hand  figure  of  each  partial  product  under 
the  figure  of  the  multiplier  used  in  obtaining  it,  and  add  the 
partial  products. 

Point  off  from  the  right  of  the  product  as  many  figures  for 
decimals,  prefixing  zeros  if  necessary,  as  there  are  decimal 
places  in  the  multiplicand  and  multiplier  together. 

Proof.  Interchange  the  multiplier  and  multiplicand,  and 
multiply.  If  the  results  agree,  the  work  may  be  assumed  to 
be  correct. 


Exercise  12. 

Find  the  product  of  : 

1. 

6  X  0.5235988. 

19. 

2.23607  X  2.236. 

2. 

4  X  0.7853982. 

20. 

0.618  X  618. 

3. 

3.14159265  X  5  X  5. 

21. 

0.618034  X  0.618035 

4. 

30  X  8.75. 

22. 

12  X  0.12936. 

5. 

0.07  X  6.975. 

'23. 

7.92801  X  0.9. 

6. 

700  X  7.81. 

24. 

58.383  X  0.39. 

7. 

8000  X  65.432. 

25. 

0.08X0.28744. 

8. 

300  X  7.85. 

26. 

0.065  X  491.205. 

9. 

0.009  X  10,356.78. 

27. 

68.325X6.25. 

10. 

7.37  X  0.785398. 

28. 

0.732  X  1.6. 

11. 

8.56  X  0.785398. 

29. 

0.438  X  1208.88. 

12. 

1001  X  0.785398. 

30. 

498  X  0.0125. 

13. 

0.083  X  2150.42. 

31. 

7  X  0.007. 

14. 

0.75  X  2150.42. 

32. 

1000  X  0.0001. 

15. 

0.075  X  2150.42. 

33. 

0.235  X  10.24. 

16. 

0.7071  X  1.4142136. 

34. 

0.00507702  X  0.0283. 

17. 

1.41421  X  1.4142. 

35. 

89.3  X  0.00752. 

18. 

1.732  X  1.732. 

36. 

74.1  X  0.0256. 

30  MULTIPLICATION. 

69.  Powers  and  Roots.  If  a  product  consists  of  equal 
factors,  it  is  called  a  power  of  that  factor  ;  and  one  of  the 
equal  factors  is  called  a  root  of  the  product. 

The  power  is  named  from  the  number  of  equal  factors. 
Thus, 

25  (5  X  5)  is  the  second  poiuer,  or  square,  of  5. 
125  (5X5X5)   is  the  third  power,  or  cube,  of  5. 
625  (5X5X5X5)  is  the  fourth  power  of  5. 

5  is  called  the  second  root,  or  square  root,  of  25. 

5  is  called  the  third  root,  or  cube  root,  of  125. 

5  is  called  the  fourth  root  of  625. 
To  avoid  writing  long  rows  of  equal  factors,  a  figure, 
called  the  exponent,  is  written  at  the  right  of  a  number 
to  show  how  many  times  the  number  is  taken  as  a  factor. 

Thus,  66  means  the  same  as5X5X5x6x6,  and  is  read  the 
fifth  power  of  6.  38  times  3*  means  3x3x3  times  3x3x3x3  = 
8X8X8X8X8X8X8  =  3*.     Hence, 

The  product  of  two  or  more  powers  of  the  same  number 
may  be  expressed  by  writing  the  number  with  an  exponent 
equal  to  the  sum  of  the  exponents  of  the  given  powers. 

Exercise  13. 
Express  the  product  of  : 

1.  76X7S;    82X8;   28  X  2  ;   54  X  52. 

2.  3.012  X  3.01  ;  0.67s  X  0.678 ;  0.208  X  0.2083. 

3.  2.0032  X  2.0034  ;  20.03*  X  20.03 ;  20.03  X  20.032- 

70.  Casting  out  Nines.  The  product  of  two  integral 
factors  is  called  a  midtiple  of  either  of  its  factors. 

Every  power  of  10  is  one  more  than  some  multiple  of  9. 
Thus,  10  =  9  +  1;  102=  11  X  9+  1;  108=  111  X  9 +1,  etc. 

Every  multiple  of  a  power  of  10  by  a  single  digit  is, 
therefore,  some  multiple  of  9,  plus  that  digit. 
Thus,  500  =  55  X  9  +  5  ;  7000  =  777  X  9  +  7,  etc. 


MULTIPLICATION.  31 

But  as  every  number  consists  of  the  sum  of  such  multi- 
ples of  powers  of  10,  every  number  is  a  multiple  of  9,  plus 
the  sum  of  its  own  digits. 

Thus,  24,573  is  a  multiple  of  9  plus  2  +  4+5  +  7  +  3.  In  casting 
out  the  nines  from  a  number,  the  remainder,  therefore,  is  the  same 
as  that  arising  from  casting  out  the  nines  from  the  sum  of  its  digits. 

In  finding  the  remainder  from  casting  out  the  nines  from 
a  number,  we  may,  of  course,  omit  the  nines,  or  any  two 
or  three  digits  which  we  see  at  a  glance  will  make  9. 

Thus,  to  cast  out  the  nines  from  1,926,754,  Ave  see  at  once  that  1, 
2,  6,  and  5,  4,  make  nines,  and  the  single  7  will  be  the  remainder. 
So  in  254,786,  we  reject  5,  4,  and  2,  7,  and  add  only  8,  6,  from  the 
sum  of  which  reject  9,  and  there  is  left  5. 

71.   Example.     Multiply  761  by  147. 
Solution.     The  remainder  after  the  nines  are  cast  out 


H  |  Multiply. 


From      761 is  5 

From     J47  .... 

5327               From    15     .  .  is  6 
3044 
761 
From  111867 ' is  6 

The  product  of  the  two  given  numbers  has  6  remaining,  and  the 
product  of  the  two  remainders  has  6  remaining  after  the  nines  are 
cast  out.     Therefore,  the  work  may  be  assumed  to  be  correct. 

72.  Casting  out  Elevens.  Even  powers  of  10  are  mul- 
tiples of  11,  plus  1  ;  odd  powers  of  10  are  multiples  of  11, 
minus  1. 

Thus,  102,  or  100  =  9  X  11  +  1  ;  104,  or  10,000  =  909  X  11  +  1,  etc. ; 
10  =  11  -  1  ;  103,  or  1000  =  91  X  11  -  1  ;  105  =  9091  X  11  -  1,  etc. 

Hence,  if  the  elevens  are  cast  out  from  a  number  ex- 
pressed by  two  digits,  the  remainder  equals  the  digit  in 
the  odd  place  minus  the  digit  in  the  even  place,  (the  digit 
in  the  odd  place,  if  less  than  that  in  the  even  place,  being 


32  MULTIPLICATION. 

increased  by  11)  ;  and,  if  the  elevens  are  cast  out  from  any 
number,  the  remainder  equals  the  sum  of  the  digits  in  the 
odd  places  (increased  by  a  multiple  of  11  if  necessary) 
minus  the  sum  of  the  digits  in  the  even  places. 

73.  The  proof  by  casting  out  elevens  is  similar  to  that 
by  casting  out  nines  ;  and  if  a  process  stands  both  tests, 
the  only  possible  error  is  a  multiple  both  of  9  and  of  11. 

Multiply  67,853  by  2976,  and  test  by  casting  out  the 
elevens. 


2976  .  . 

q  y  Multi 

407118 

30  ...  . 

474971 

610677 

135706 

201930528.  .  . 

74.  In  applying  either  test  to  decimals,  disregard  the 
decimal  point.  In  questions  involving  large  numbers,  al- 
ways apply  one  or  both  tests  to  the  work. 

Exercise  14. 

Find  the  following  products,  and  test  the  accuracy  by 
casting  out  the  nines  and  by  casting  out  the  elevens : 

1.  21.3706  X  15.243  X  1.8954. 

2.  0.026891  X  5.328  X  29.74. 

3.  0.0012  X  5.8281  X  0.6827. 

4.  23.9875  X  12.4764  X  0.017. 

5.  39.801  X  1.44  X  17.9645. 

6.  0.0165  X  5.2817  X  0.8469. 

7.  0.54237  X  16  X  0.00176. 

8.  24.271  X  3.6485  X  15.271. 

9.  13.256  X  14.125  X  30.254. 


MULTIPLICATION. 


33 


75.  Contracted  Multiplication  of  Decimals.  In  ordi- 
nary calculations  we  seldom  use  a  decimal  smaller  than 
0.00001  of  the  unit. 

Multiply  0.123456789  by  1.23456789. 


6 

49 

370 

2469 

12345 


0.123456789 
1.23456789 
1111111101 
987654312 
864197523 

740740734 

17283945 

3827156 

370367 

13578 

6789 


0.15241578750190521 


The  6  at  the  left  of  the  vertical  line  is 
obtained  by  multiplying  the  1  in  the  mul- 
tiplicand by  5  in  the  multiplier,  and  carry- 
ing 1  from  5X2.  The  9  below  the  6  is 
obtained  by  multiplying  the  2  by  4,  and 
carrying  1  from  4x3.  The  0  below  the 
9  is  obtained  by  multiplying  the  3  by  3,  and 
carrying  1  from  3X4.  The  9  below  the  0 
is  obtained  by  multiplying  the  4  by  2,  and 
carrying  1  from  2X5.  The  5  below  the 
9  is  obtained  by  multiplying  the  5  by  1. 


It  is  evident  that,  if  five  decimal  places  only  are  required 
in  the  product,  all  the  work  to  the  right  of  the  vertical  line 
is  wasted. 

To  shorten  the  work,  disregard  the  decimal  point  in 
the  multiplier,  and  write  the  multiplier  under  the  multi- 
plicand, so  that  each  digit  of  the  multiplier  shall  fall 
directly  under  the  digit  of  the  multiplicand  into  which  it 
is  multiplied  to  produce  the  first  figure  to  the  left  of  the 
vertical  line  in  each  partial  product.     Thus  : 


0.123456789 
987654321 
12346 


370 

49 

6 

1 


0.15241 


Notice  that  the  figures  of  the  multiplier  are  re- 
versed ;  that  the  units'  figure  of  the  multiplier 
falls  under  the  last  decimal  of  the  multiplicand 
which  is  to  be  retained,  and  that  the  decimals  in 
the  result  are  correct.  In  order,  however,  to 
have  the  required  number  of  decimal  places  cor- 
rect, it  will  generally  be  necessary  to  take  one 
more  than  the  required  number  of  decimal 
places.    Hence, 


34  MULTIPLICATION. 

76.   To  Multiply  Decimals  by  the  Contracted  Method, 

Reverse  the  multiplier,  and  put  the  units?  figure  under  the 
last  place  of  decimals  to  be  retained. 

Multiply  each  figure  of  the  multiplier  into  the  figure  next 
to  the  right  above  it.  Do  not  write  this  result,  but  carry  the 
nearest  ten  to  the  next  result,  multiplying  as  usual. 

Write  the  first  figures  of  the  partial  products  in  a  vertical 
column. 

Add  the  products,  and  point  off  from  the  sum  as  many 
decimal  places  as  were  taken  in  the  multiplicand. 

If  the  multiplier  has  no  units'  figure,  supply  its  place  with 
a  zero. 

To  insure  accuracy,  take  one  decimal  place  more  than  the 
required  number. 

To  detect  errors  that  may  arise  from  displacement  of  the 
decimal  point,  or  from  an  erroneous  arrangement  of  the  fac- 
tors, test  the  result  by  a  rough  estimate  of  what  the  product 
should  be. 

Exercise  15. 
Find  to  the  fifth  decimal  the  value  of : 

1.  0.49714987X1.75812263. 

2.  0.79817987  X  0.99429975. 

3.  1.09920986  X  0.24857494. 

4.  0.62208861  X  0.16571662. 

5.  1.75812263  X  2.05915963. 

6.  0.55630251  X  0.33445375. 

7.  0.75142506  X  9.98998569. 

8.  0.05245506  X  0.16571662. 

9.  0.33143325  X  1.79263713. 

10.  0.90633287X0.6154551376. 

11.  2.84657842  X  0.96695542. 

12.  0.546794489X2.847697495. 


CHAPTER   IV. 
DIVISION. 

77.  Division.  If  the  product  and  one  factor  are  given, 
the  process  of  finding  the  other  factor  is  called  division. 

78.  Dividend.  The  given  product,  that  is,  the  number  to 
be  divided,  is  called  the  dividend. 

79.  Divisor.  The  given  factor,  that  is,  the  number  by 
which  the  dividend  is  divided,  is  called  the  divisor. 

80.  Quotient.  The  required  factor,  that  is,  the  number 
found  by  division,  is  called  the  quotient. 

81.  To  divide  35  apples  by  7  apples  is  to  find  the  number 
of  times  we  must  take  7„  apples  to  obtain  35  apples.    Hence, 

If  the  divisor  and  dividend  denote  the  same  kind  of  units, 
the  quotient  is  an  abstract  number. 

82.  To  divide  35  apples  by  7  is  to  find  the  number  of 
apples  in  each  part,  when  35  apples  are  divided  into  7  equal 
parts.     Hence, 

If  the  divisor  is  an  abstract  number,  the  quotient  denotes 
units  of  the  same  kind  as  the  dividend. 

83.  What  is  one  of  the  parts  called,  if  a  number  is 
divided  into  2  equal  parts  ?3?4?5?6?7?8?9? 

One  half  is  written  \  ;  one  third,  -J- ;  one  fourth,  \  ;  one 
fifth,  \-,  one  sixth,  \\  one  seventh,  \;  one  eighth,  £;  one 
ninth,  \;  and  so  on. 

To  divide  35  apples  by  7  is  to  find  \  of  35  apples. 


36  DIVISION. 

84.  Division  is  indicated  by  the  sign  of  division  -r-,  or  by 
writing  the  dividend  over  the  divisor  with  a  line  between 
them. 

Thus,  42  -j-  6  and  -\2  have  the  same  meaning,  and  each  is  read  : 
forty-two  divided  by  six. 

85.  Remainder.  If  the  divisor  does  not  exactly 
divide  the  dividend,  the  part  of  the  dividend  left  from  the 
division  is  called  the  remainder. 

Thus,  $42  t4=  $10  with  remainder  |2. 

86.  Principles  of  Division.  The  value  of  the  quotient 
depends  upon  the  relative  values  of  the  dividend  and 
divisor. 

Suppose  we  have  36  -f-  6  =  6. 

If  we  multiply  the  dividend  36  by  2,  what  effect  will  this  have  on 
the  quotient  ? 

If  we  divide  the  dividend  36  by  2,  what  effect  will  this  have  on  the 
quotient  ? 

If  we  multiply  the  divisor  6  by  2,  what  effect  will  this  have  on  the 
quotient  ? 

If  we  divide  the  divisor  6  by  2,  what  effect  will  this  have  on  the 
quotient  ? 

If  we  multiply  both  the  dividend  and  divisor  by  2,  what  effect  will 
this  have  on  the  quotient  ? 

If  we  divide  both  the  dividend  and  divisor  by  2,  what  effect  will 
this  have  on  the  quotient  ? 

From  the  answers  to  these  questions,  we  have  the  follow- 
ing important  principles  of  division  : 

Multiplying  the  dividend  or  dividing  the  divisor  by  a 
number  multiplies  the  quotient  by  that  number. 

Dividing  the  dividend  or  multiplying  the  divisor  by  a 
number  divides  the  quotient  by  that  number. 

Multiplying  both  dividend  and  divisor  by  the  same  number, 
or  dividing  both  by  the  same  number,  does  not  change  the 
quotient. 


DIVISION.  37 

Short  Division. 

87.  If  the  divisor  is  so  small  that  the  work  can  be  per- 
formed mentally,  the  process  is  called  short  division. 

88.  Examples.     1.   Divide  976  by  3. 

Solution.     We  write  the  divisor  to  the  left  of  the  dividend  with  a 

curved  line  between  them,  and  the  quotient  under  the  dividend,  as  in 

the  margin.     3  is  contained  in  9  three  times  ;  and  we  write 

3)976       3  in  the  quotient  under  the  9  of  the  dividend.     3  is  con- 

325£    tained  in  7  twice  with  remainder  1  ;  and  we  write  2  in  the 

quotient  in  the  place  of  tens.     The  remainder  1  is  1  ten 

or  10  units,  and  10  units  +  the  6  units  of  the  dividend  makes  16  units. 

3  is  contained  in  16  five  times  with  remainder  1.     The  answer  is  325 

with  remainder  1.    We  may  write  the  remainder  over  the  divisor  as  a 

part  of  the  quotient;  thus,  325£. 

The  following  wording  and  no  more  should  be  used  :  3  in  9,  3  ;  in 
7,  2 ;  in  16,  5  remainder  1. 

2.  Divide  72.56  by  8. 

Solution.     8  in  72,  9  ;  in  5,  0  ;  in  56,  7.   Here  the  72  to  be  divided 

by  8  is  72  units,  and  72  units  divided  by  8  gives  9  units  ; 

8)72.56    we  place  the  decimal  point  in  the  quotient  after  the  units' 

9.07     figure  9.     The  next  figure  5  is  5  tenths,  and  5  will  not 

contain  8.     We  write  0  in  the  quotient  and  annex  the  6 

hundredths  to  the  5  tenths,  making  56  hundredths  ;  and  56  hundredths 

divided  by  8  is  7  hundredths.     The  quotient  is  9.07.     Hence, 

If  the  divisor  is  an  integral  number,  we  write  the  decimal 
point  in  the  quotient  as  soon  as  we  reach  the  decimal  point  in 
the  dividend. 

3.  Divide  72.56  by  0.08. 

Solution.  Since  we  do  not  change  the  quotient  if  we  multiply 
both  divisor  and  dividend  by  the  same  number  (§  86),  we 

8)7256  make  the  divisor  an  integral  number  by  multiplying  it  by 
907  100,  and  multiply  the  dividend  by  the  same  number. 
We  then  divide  as  before.     Hence, 

If  the  divisor  contains  decimal  places,  we  multiply  both 
divisor  and  dividend  by  10,  or  some  power  of  10,  so  as  to 
make  the  divisor  an  integral  number. 


38  DIVISION. 

4.   Divide  78.52  by  8000. 

Solution.     We  first  divide  the  divisor  by  1000  by  cutting  off  the 

three  zeros  at  its  right ;    and  move  the  decimal 

8ffff0)O.O7852      point  in  the  dividend  three  places  to  the  left,  pre- 

0.009815    fixing  one  zero.     When  we  reach  the  last  figure 

of  the  dividend,  we  mentally  supply  a  zero,  and 

continue  dividing.     Hence, 

If  the  divisor  is  an  integral  number  ending  in  one  or  more 
zeros,  we  cut  off  the  zeros,  and  mave  the  decimal  point  in  the 
dividend  as  many  places  to  the  left  as  we  cut  off  zeros. 

89.  If  we  add  the  remainder  to  the  product  of  the  divisor 
and  quotient,  we  obtain  the  dividend.     Hence, 

We  find  the  product  of  the  divisor  and  quotient,  and  to  tl>  it 
prod  mi  add  tJie  remainder.  If  the  result  equals  the  dividend, 
the  work  may  be  assumed  to  be  correct. 

Exercise  16. 

Find  the  quotient  of : 

1.  126.409-^9.  15.  87,585 -^  1200. 

2.  13.31 -M0.  16.  27,485 -f- 200. 

3.  13.31 -f- 11.  17.  10.01-^0.02. 

4.  1.728  -T-12.  18.  0.04^0.05. 

5.  37.632-^30.  19.  7.432 -f- 0.04. 

6.  42,631-^20.  20.  31-^-0.005. 

7.  96,464 -MOO.  21.  480-^0.012. 

8.  58.775-^-600.  22.  980^-0.0007. 

9.  75,230 -M00.  23.  10.98^-0.00009. 

10.  8956-^80.  24.  10.98^0.09. 

11.  98,254-^900.  25.  0.1098-^0.00009. 

12.  82,610 -^  7000.  26.  0.1098 -f- 0.09. 

13.  83,690-!- 500.  27.  1441  -4-  0.11. 

14.  96,464 -M 10.  28.  18.92  -M.l. 


DIVISION.  39 

Long  Division. 

90.  The  process  of  Long  Division  is  the  same  as  that 
of  Short  Division,  except  that  the  work  is  written  in  full, 
and  the  quotient  is  written  over  the  dividend,  the  first 
quotient  iigure  being  written  over  the  right-hand  figure  of 
the  partial  dividend  used  in  obtaining  it. 

Note.  The  quotient  may  be  written  at  the  right  of  the  dividend. 
The  advantage,  however,  of  writing  the  quotient  over  the  dividend  is 
easily  seen  when  the  quotient  contains  a  decimal. 

91.  Each  quotient  figure  may  be  estimated  by  taking  for 
a  trial  divisor  the  nearest  number  of  tens  or  hundreds, 
etc.,  represented  by  the  divisor,  and  by  taking  for  a  trial 
dividend  the  nearest  number  of  tens  or  hundreds,  etc., 
represented  by  the  partial  dividend. 

92.  In  each  step  of  division,  the  product  must  be  less  than 
the  partial  dividend,  and  the  remainder  less  than  the  divisor. 

93.  Examples.     1.,  Divide  4199  by  78. 

Solution.     As  78  is  more  than  41,  it  is  necessary  to  take  three 
figures  of  the  dividend  for  the  first  partial  dividend.     As  the  nearest 
number  of  tens  represented  by  the  divisor 
&3  is  8,    we  take  8   for  the  trial  divisor.     As 

78)4199  the  nearest  number  of   tens  represented  by 

390  the  partial  dividend  is  42,  we  take  42  for  the 

299  trial  dividend.      8   is  contained   5  times  in 

234  42.     Hence,  the  first   quotient   figure    is   5, 

65  remainder,  and  we  write  5  in  the  quotient  over  the  right- 
hand  figure,  9,  of  the  partial  dividend.  We 
multiply  the  divisor  78  by  5  and  subtract  the  product  390  from  419. 
We  annex  the  next  figure  9  of  the  dividend  to  the  remainder  29. 
The  nearest  number  of  tens  represented  by  the  second  partial  divi- 
dend is  30,  and  8  is  contained  3  times  in  30.  We  place  3  as  the 
second  figure  of  the  quotient  and  multiply  the  divisor  by  3.  This 
product  subtracted  from  the  second  partial  dividend  leaves  for  a 
remainder  65.     The  complete  quotient  may  be  written  53^|. 


40  DIVISION. 

2.    Divide  2791.163  by  394. 

Solution.     The  first  partial  dividend  is  2791.   As  the  nearest  num- 
ber of  hundreds  represented  by  the  divisor  394  is  4,  we  take  4  for  the 
trial  divisor.     As  the  nearest  number  of  hundreds  represented  by  the 
partial  dividend  2791  is  28,  we  take  28 
7.084  for  the  trial  dividend.     4  is  contained 

394)2791.163  7  times  in  28.     We  write  the  7  over  the 

2758  1,   the  right-hand  figure  of  the  partial 

3316  dividend.     We  place  the  decimal  point 

31. "i2  in  the  quotient  directly  over  the  decimal 

1643  point  in  the  dividend,  that  is,  directly 

1576  after  the   7.     We   multiply  the  divisor 

67  remainder.  394  by  7.  We  subtract  the  product  2758 
from  2791  and  have  for  a  remainder  33, 
to  which  we  annex  the  1  of  the  dividend.  As  331  is  less  than  394, 
the  next  quotient  figure  is  0.  To  331  we  annex  the  next  figure,  6,  of 
the  dividend.  4  is  contained  8  times  in  33.  We  therefore  write  8 
for  the  next  quotient  figure,  and  find  the  product  of  8  X  394  to  be 
3162.  The  remainder  obtained  by  subtracting  3152  from  3316  is  164, 
to  which  we  annex  the  3  of  the  dividend.  4  is  contained  4  times  in 
16.  The  product  of  4  X  394  is  1576,  and  this  subtracted  from  1643 
leaves  67  for  the  final  remainder.  The  complete  quotient  may  be 
written  7.084^*. 

17  When  the  quotient  contains  a  decimal,  it  is  not  cus- 
394^670  tomary  to  write  the  remainder  over  the  divisor,  but 
qo4  to  continue  the  division  by  annexing  zeros  to  the  divi- 
^g^  dend.  In  this  example,  if  we  carry  the  division  two 
2758  decimal  Peaces  further,  as  in  the  margin,  we  have  the 
quotient  7.08417. 

94.  Decimals  are  seldom  carried  to  more  than  five 
places.  If  we  wish  to  find  the  value  of  a  decimal  correct 
to  the  nearest  tenth,  hundredth,  thousandth,  etc.,  we  add  1 
to  the  last  required  figure  if  the  next  figure  would  be  five 
or  more. 

Thus,  the  value  of  the  answer  to  the  last  example,  7.08417,  correct 
to  the  nearest  tenth  is  7.1 ;  correct  to  the  nearest  hundredth  is  7.08  ; 
correct  to  the  nearest  thousandth  is  7.084  ;  correct  to  the  nearest  ten- 
thousandth  is  7.0842. 


DIVISION.  41 

95.  As  in  Short  Division,  if  the  divisor  contains  deci- 
mal places,  we  multiply  both  divisor  and  dividend  by  a 
power  of  10,  so  as  to  make  the  divisor  an  integral  number. 

Examples.     1.   Divide  28.3696  by  1.49. 

(1)  (2) 

19.04  19.04 


149)2836.96  1.49)28.36^96 

149  149 

1346  1346 

1341  1341 

596  596 

596  596 

In  form  (1)  we  multiply  both  dividend  and  divisor  by  100. 

The  same  result  is  obtained  by  counting  to  the  right  from  the 
decimal  point  in  the  dividend  as  many  places  as  there  are  decimal 
places  in  the  divisor,  inserting  a  caret  as  shown  in  form  (2),  and 
putting  the  decimal  point  in  the  quotient  directly  over  the  caret. 

2.   Divide  0.381876  Jay  2.63;  739.4112  by  0.1728. 


0.1452 

4279. 

2.63)0.38A1876 

0.1728)739.4112A 

263 

6912 

1188 

4821 

1052 

3456 

1367 

13651 

1315 

12096 

526 

15552 

526 

15552 

3.   Divide  34.2  by  18,000. 

0.0019 
180^0) A034. 2        Solution.     Here  we  cut  off  the  three  zeros  from 

18  the  divisor,  and  put  a  caret  three  places  to  the  left 

162  of  the  decimal  point  in  the  dividend  ;  that  is,  we 

162  divide  both  the  divisor  and  the  dividend  by  1000. 


42  DIVISION. 

96.   From  these  examples  we  have  the  following 

Rule  fob  Long  Division.  Write  the  divisor  to  the  left 
of  the  dividend,  with  a  curved  line  between  them. 

If  the  divisor  contains  decimal  places,  remove  the  decimal 
point  from  the  divisor,  and  move  the  decimal  point  in  the 
dividend  to  the  right  as  many  places  as  there  are  decimal 
jjlaces  in  the  divisor. 

Take  for  the  first  partial  dividend  the  fewest  left-hand 
figures  thai  will  contain  the  divisor,  <nnl  write  the  quotient 
over  the  right-hand  figure  of  this  partial  dividend. 

Multiply  the  divisor  by  tliis  quotient,  and  place  the  product 
under  the  partial  dividend  used. 

Subtract  this  product,  and  to  the  remainder  annex  the  next 
figure  of  the  dividend. 

Proceed  as  before,  and  continue  the  process  until  all  the 
figures  of  the  dividend  have  been  used  ;  putting  the  decimal 
jmint  in  the  quotient  as  soon  as  the  decimal  point  in  the 
dividend  is  reached. 

Proof.  Find  the  product  of  the  divisor  and  quotient,  and 
to  this  j  >r<><l  net  a  (hi  the  remainder,  if  any.  If  the  result  equals 
the  dividend,  the  work  mag  be  assumed  to  be  correct. 

Note.  If  the  divisor  is  an  integral  number  ending  in  one  or  more 
zeros,  cut  off  the  zeros,  and  move  the  decimal  point  in  the  dividend 
to  the  left  as  many  places  as  there  are  zeros  cut  off,  prefixing  zeros 
if  necessary. 

Exercise  17. 
Find  the  quotient  of  : 

1.  7553 -f- 91.  5.  35,372-^35.  9.  370,406 -r- 843. 

2.  45934-73.  6.  834,561-^408.  10.  978.217 -M98. 

3.  89,713  4-76.  7.  341,586 -f- 247.  11.  543,816  4-357. 

4.  53,691^-88.  8.  861,3454-395.  12.  604,730  4-1289. 


DtVlSlOtt. 

13. 

326.7  4-132. 

47. 

125,457.64  4-  354.2. 

14. 

79.72552  4- 1.121. 

48. 

0.75-^0.866. 

15: 

0.4077 -f- 9.06. 

49. 

96,800  4-311.13. 

16. 

961.9476  4-106.8. 

50. 

2,534,000-1-6.4037. 

17. 

$27,121.50  4- $387.45. 

51. 

6.4037  4-2,534,000. 

18. 

123.684792  4-  39.37. 

52. 

0.7407^5.504. 

19. 

1203.75  4-19.26. 

53. 

54.44  4-  1.7359. 

20. 

256  4-0.0016. 

54. 

0.3658  4-2.322. 

21. 

24.1802  4-3.19. 

55. 

$25,12  4-1.43. 

22. 

200-^0.3125. 

56. 

17-^0.036. 

23. 

7.704256  -i-  0.08302. 

57. 

$143  4-1.7892. 

24. 

1.6093295  4-0.479. 

58. 

18.7  4-121. 

25. 

1.6093295^-0.917. 

59. 

495,872.17654-1728. 

26. 

1.6093295^-0.017. 

60. 

186,517.725  4-5280. 

27. 

1.6093295  4-0.0087. 

61. 

56,287.625  4-231. 

28. 

3-^1.7. 

62. 

782,847.375  4-43,560, 

29. 

3-^1.73. 

63. 

18,520  4-272.25. 

30. 

34-1.732. 

64. 

36,581.17-^2150.42. 

31. 

3  4-1.7321. 

65. 

10,000  4- 196. 

32. 

1.6093295  4-5280. 

66. 

$219.12 -M.025. 

33. 

2  4-  1.4142. 

67. 

22,000-^5645.376. 

34. 

54-2.236. 

68. 

0.0165  4-1.331. 

35. 

$25,000  4-117. 

69. 

75,555  4-1152. 

36. 

162-^14.72. 

70. 

12.2  4-  5.5056. 

37. 

1.27  4-19,800. 

71. 

77^-10.7716. 

38. 

0.5632-^1.6382. 

72. 

66  4-7.2426. 

39. 

1,872,760  4-  42,360. 

73. 

54.55  4- 1728. 

40. 

1897  4-192.93. 

74. 

46.88  4-44.723. 

41. 

0.00014-0.0872. 

75. 

0.874  4-444. 

42. 

14.1658  4- 1.8246. 

76. 

54,728  4-  5280. 

43. 

$10,150.75  4- $303.77. 

77. 

2.034  4-  0.0018. 

44. 

$312.12-^24.73. 

78. 

0.08748 -^  10.8. 

45. 

$3115.20-4176. 

79. 

4443.33  4-  0.0037. 

46. 

$2840-^5.135. 

80. 

0.032048  4-20.03. 

48 


44  DIVISION. 

97.  The  Parenthesis.  If  numbers  are  included  in  a 
parenthesis  (  ),  the  first  step  is  to  reduce  these  numbers  to 
a  single  number  as  the  signs  direct. 

(9+ 7 -1)-^  5=  15-^5.  (4  X  6-9) -r  5=15-^5. 

48  -r  (4  X  3)  =  48  4- 12.  48  -r  (24  -r  4)  =  48  +  6. 

Note.  Instead  of  a  parenthesis  we  sometimes  use  brackets  [], 
braces  {  },  or  a  vinculum  — .  Thus,  (8-3),  [8-3],  {  8  -  3 }, 
8  —  3,  all  have  the  same  meaning. 

In  reducing  expressions  containing  the  signs  +,  — ,  X, 
-7-,  we  first  perform  the  operations  indicated  by  the  signs 
X  and  +  in  the  order  in  which  they  stand  ;  then  the  opera- 
tions indicated  by  +  an(i  — • 

Thus,  48-r8X2-  3X2  +  GX  6  -r  2  =  12  -  6  +  15  =  21. 


Exercise  18. 
Reduce  to  a  single  expression  : 

1.  (16-11  +  2)  X  5.  3.    (84  +  7)  +  (4  +  5-6). 

2.  (4  X  15)  +  (2  X  3).         4.    (44-31)  X  (14 -11). 

5.  (96+6  +  5)-(6x8  +  16). 

6.  (52-5X7) +  (4X5) -16  +  2. 

7.  52-5X7  +  4X5-16  +  2. 

8.  (62  +  3-15)  +  10  +  (6X7-30)  +  3. 

98.  Cancellation.  The  work  of  division  may  often  be 
shortened  by  dividing  out,  or  cancelling,  equal  factors  from 
the  divisor  and  dividend. 

Divide  1596  by  84. 

Solution.     It  is  readily  seen  that  12  is  contained  in  84 
12)1596    and  in  1596.     Dividing  each  by  12,  we  have  7  and  133  ; 
7)133    and  we  find  the  quotient  19  by  dividing  133  by  7  by 
19     short  division. 


DIVISION.  45 

••  99*.  Reciprocals.     If  the  product  of  two  numbers  is  1, 
each  of  the  numbers  is  called  the  reciprocal  of  the  other. 

Thus,  2  X  0.5  =  1  ;  hence,  2  is  the  reciprocal  of  0.5,  and  0.5  is  the 
reciprocal  of  2.  Again,  0.8  X  1.25  =  1 ;  hence,  0.8  is  the  reciprocal  of 
1.25,  and  1.25  is  the  reciprocal  of  0.8.  So  4  is  the  reciprocal  of  0.25  ; 
3  is  the  reciprocal  of  0.33333  ;  and  so  on. 

100.  When  the  dividend  is  1,  the  quotient  is  the  recipro- 
cal of  the  divisor;  and  when  the  dividend  is  any  other 
number  than  1,  the  quotient  is  the  reciprocal  of  the  divisor 
multiplied  by  that  number.     Hence, 

101.  To  divide  a  number  by  a  divisor  gives  the  same  result 
as  to  multiply  the  number  by  the  reciprocal  of  the  divisor. 

Also,  to  multiply  a  number  by  a  multiplier  gives  the  same 
result  as  to  divide  the  number  by  the  reciprocal  of  the  multi- 
plier. 

102.  The  processes  of  multiplication  and  division  are 
often  made  much  simpler  by  using  the  reverse  process  with 
the  reciprocal  of  the  multiplier  or  of  the  divisor. 

Thus,  to  divide  2.71827  by  37.5,  since  37.5  =  3  X  12.5,  we  may 
divide  2.71827  by  3,  and  multiply  the  quotient  by  0.08. 

Exercise  19. 

By  the  use  of  reciprocals,  find  the  value  of : 

1.  8X0.25.  10.  1764  X  0.025. 

2.  171-^0.25.  11.  5381-^-0.025. 

3.  876X1.25.  12.  7452-^0.875. 

4.  132X2.5.  13.  651X0.33333. 

5.  591 -f- 2.5.  14.  456X6.66667. 

6.  756-^0.125.  15.  1554X0.16667. 

7.  268X25.  16.  432^-1.33333. 

8.  753-^25.  17.  375  -=-16.66667. 

9.  567-^625.  18.  225-^6.66667. 


46  DIVISION. 

103.  Contracted  Division  of  Decimals.  Annexing  a 
digit  to  a  partial  dividend  multiplies  the  partial  dividend 
by  10  and  adds  to  the  product  the  number  expressed  by 
the  digit. 

Cutting  off  a  digit  from  the  right  of  the  divisor  subtracts 
from  the  divisor  the  number  expressed  by  the  digit  and 
divides  the  remainder  by  10.  The  quotient  figure,  there- 
fore, will  be  the  same  whichever  we  do  ;  and  this  principle 
can  be  applied  to  shorten  the  labor  of  division  in  examples 
involving  long  decimals. 

Divide  15.4323487  by  1.4142136,  to  four  decimal  places. 

10*9123  it  is  necessary  to  determine,  first, 

14142136)154323487.  the  number  of  significant  figures 

14142136  required  in  the  quotient. 

129021270  Since  1.4  is  contained  in  15  ten 

127279224  times,  it  is  evident  that  the  quo- 

17420460  tient  will  have  two  integral  places. 

14142136  The  two  integral  figures  and  the 

32783240  four  decimal  figures  make  six,  which 

28284272  will  be  the  number  of  figures  re- 

44989680  quired  in  the  quotient. 
42426408 

104.  In  general,  to  determine  the  number  of  significant 
figures  required  in  the  quotient,  move  the  decimal  point 
of  the  divisor  to  the  right  of  the  first  significant  figure, 
and  move  the  decimal  point  of  the  dividend  as  many 
places,  and  in  the  same  direction,  as  the  decimal  point  of 
the  divisor  has  been  moved.  The  number  of  integral  places 
in  the  quotient  can  then  easily  be  determined. 

It  is  best  not  to  cut  off  figures  from  the  right  of  the 
divisor  until  the  number  of  figures  still  required  in  the  quo- 
tient is  two  less  than  the  number  of  digits  in  the  divisor. 
In  multiplying  the  divisor  by  each  quotient  figure,  multiply 
the  figure  of  the  divisor  cut  off,  and  carry  the  nearest  ten. 


DIVISION.  47 

The  work  may  be  arranged  as  follows  : 

Cut  off  the  6.     The  first  product  is  increased  by  1  for  the  1  X  6 
omitted.      The   first    remainder   is    in- 

10-9123  creased  by  1  for  the  8  in  the  dividend. 

UlfflXW)  154323487.        "  Cut  off  the  3.     As  the   divisor  is  not 

1414214  contained  in  the  partial  dividend,  we 

129021  also  cut  off  the  1.     The  product  by  9 

127279  is  increased  by  1  for  the  9  X  1  omitted. 

1742  Cut  off  the  2.     As  1  X  2  is  less  than  5, 

1414  the  product  by  1  is  not  increased.     Cut 

328  off  the  4.      The   product   by  2  is  in- 

283  creased  by  1  for  2  X  4  omitted.     Cut  off 

45  the    1.      The   product  by  3  is  not  in- 

42  creased,  for  3  X  1  is  less  than  5.    Hence, 

105.    To  Divide  Decimals  by  the  Contracted  Method, 

Determine  the  number  of  significant  figures  required  in 
the  quotient.  Begin  to  cut  off  figures  from  the  right  of  the 
divisor,  when  the  number  of  figures  still  required  in  the  quo- 
tient is  two  less  than  the  number  of  digits  in  the  divisor.  In 
multiplying  the  divisor  by  each  quotient  figure,  multiply  the 
figure  of  the  divisor  cut  off,  carrying  the  nearest  ten. 

Exercise  20. 

Divide  by  the  contracted  method : 

1.  11.4285285  by  3.1415927  to  six  decimal  places. 

2.  0.004239239  by  3.2783278  to  five  decimal  places. 

3.  437  by  215.253  to  five  decimal  places. 

4.  0.0053  by  72.654  to  eight  decimal  places. 

5.  6  by  0.1573  to  three  decimal  places. 

6.  0.11  by  1937.43  to  eight  decimal  places. 

7.  44.2  by  0.768547  to  five  decimal  places. 

8.  0.6587465  by  0.5475869  to  five  decimal  places. 

9.  46  by  0.00751515151  to  three  decimal  places. 


48  DIVISION. 

106.  Equations.  A  statement  that  two  expressions  of 
number  have  the  same  value  is  called  an  equation. 

Every  equation  consists  of  two  expressions  of  number 
connected  by  the  sign  of  equality,  = ;  the  two  expressions 
are  called  the  sides  or  members  of  the  equation. 

Thus,  2  X  4  =  8,  and  G  +  4  +  5  =  18  —  3  are  equations. 

107.  Since  the  two  members  of  an  equation  are  equal, 
if  both  members  are  increased  by,  diminished  by,  multiplied 
by,  or  divided  by  equal  numbers,  the  results  are  equal. 

108.  Division  of  Powers  of  the  Same  Number. 

SinCe       f-5X5X5X5X5=5x5  = 
5*  5X5X5 

5»  5X5X5  1    ..       . 

and  5'  =  5X5X5x5l^5  =  5',theref0re' 

If  the  exponent  of  the  power  in  the  dividend  is  greater 
than  the  exponent  of  the  power  in  the  divisor, 

The  quotient  is  that  number  with  an  exponent  equal  to  the 
exponent  of  the  dividend  minus  that  of  the  divisor. 

If  the  exponent  of  the  power  in  the  divisor  is  greater 
than  the  exponent  of  the  power  in  the  dividend, 

The  quotient  is  one  divided  by  the  number  with  an  exponent 
equal  to  the  exponent  of  the  divisor  minus  that  of  the  dividend. 

Exercise  21. 

Express  the  value  of  : 

1.  101;  102;  108;  104;  105;  106;  107;  108. 

2.  103-^102;  108-M0«;  105-M08;  109-f-104. 

3.  9.994-f-9.992;  9.99108-^9.99110;  9.9916-^9.9918. 

4.  1.012* -7-1.01";  1.01M-5-1.01u;  1.01u-r-1.01M. 


DIVISION.  49 

109.   Proof  of  Division  by  Casting  out  Nines.     Divide 
1,348,708  by  498. 

If  we  divide  1,348,708  by  498,  we  have  for  quotient  2708  and  for 
remainder  124. 

That  is,  1,348,708  =  498  X  2708+124. 

But,  §  70,  1,348,708  =  a  number  of  nines  +  4, 

498  X  2708  =  a  number  of  nines  +  6, 
and  124  =  a  number  of  nines  +  7. 

Hence,  498  X  2708  +  124  =  a  number  of  nines  +  6+7 

=  a  number  of  nines  +  4. 

Therefore,  the  two  members  of  the  equation  1,348,708  =  498  X 
2708  +  124,  when  divided  by  9,  give  the  same  remainder  4. 
We  may  arrange  the  work  as  follows  : 

(    8 2708 

Multiplyj    g  498)1348708 4 


Add 


6  ....  24  996 

3527 

3486 


4108 
3984 

_7 v 124 

13 '. 


In  examples  involving  large  numbers  always  apply  this 
test. 

Exercise  22. 

Find  the  following  quotients,  and  test  the  accuracy  of 
the  work  by  casting  out  the  nines  : 

1.  157,056.41692-+ 2150.42. 

2.  49,652.7896 -+5645.376. 

3.  2000 -+3.1416. 

4.  18.97657 -+0.7854. 

5.  28,762.762 +- 14.39874. 

6.  8747.119  -+6.58298. 

7.  28,752.1 -+149.796. 


50  DIVISION. 

Exercise  23.  —  Review. 
Express  in  words  : 

1.  327.244.  3.   0.390012.  5.   0.0000008. 

2.  80.9056.  4.   20,000.002.  6.   41.27105. 

Write  in  figures  : 

7.  Two  hundred  thirty-five  and  eight  hundred  thirty-five 

thousandths. 

8.  Seventy-four  and  two  hundred  three  thousand  six  mil- 

lionths. 

9.  Twelve  hundred  and   eight  thousand  three   ten-mil- 

lionths. 

10.  Five  thousand  sixty-four  niillionths. 

11.  One  million  and  four  tenths. 

12.  Six  hundivd-millionths. 

Multiply  and  divide  : 

13.  789.365  by  10  ;  by  100 ;  by  100,000. 

14.  0.004  by  100  ;  by  10,000  ;  by  1000. 

15.  436  by  1,000,000  ;  by  1000  ;  by  10. 

16.  0.1  by  ten  ;  by  ten  millions. 

Find  the  value  of  : 

17.  21.3706  +  15.243  +  1.8954  +  0.026891+5.328  +  20.74. 

18.  57  +  0.0057  +  6.8  +  1200  +  0.847  +  159.2  +  3. 

1 9.  0.0012+  10  +  5.8281  +  5  +  39.43  +  0.6827  + 1. 

20.  23.9875  - 12.4764  ;  35.14732  -  27.62815. 

21.  102.1274  -  83.072  ;  39.801  -  17.9645. 

22.  30  — 5.2817;  1.7  — 0.8469. 

23.  1-0.54237  ;  100  —  0.00176. 

24.  24.271  —  3.6485  +  15.271  —  13.256  -  14.125. 

25.  52  +  0.52  — 17.8946  —  30.254  —  0.5  +  21.12. 


DIVISION.  51 

26.  41.289  X  0.5  ;  0.268  X  0.9  ;  0.112  X  0.2. 

27.  2.435  X  4.23  ;  71.651  X  3.37  ;  0.251  X  0.04. 

28.  0.0012  X  0.005  ;  2.26823  X  200  ;  5.6125  X  0.0768. 

29.  0.7  X  7  X  0.07  ;  0.15625  X  23.7  X  0.00192  X  5. 

30.  (2.465  + 1.21)  X  (3.2  -  2.89). 

31.  (3.01)2;  (0.045)2;  (0.0081)2 ;  (5.1004)3;  (0.76)3. 

32.  (0.125)2  X  (0.32)3. 

Divide  : 

33.  291.84  by  6  ;  0.12936  by  12  ;  7.92801  by  0.9. 

34.  58.383  by  0.39  ;  0.28744  by  0.08  ;  491.205  by  0.065. 

35.  68.325  by  6.25  ;  0.732  by  1.6;  1208.88  by  0.438. 

36.  498  by  0.0125  ;  7  by  0.007  ;  1000  by  0.0001. 

37.  0.235  by  10.24  ;  27  by  12  ;  0.00507702  by  0.0283. 

38.  89.3  by  0.00752  ;  74.1  by  0.0256  ;  1  by  0.128. 

39.  0.39842  by  3.7164  ;  281.5  by  13.789  ;  0.0005  by  0.0028. 

40.  63.04128  by  912.85  ;  287.209  by  0.00493  ;  2000  by 

0.0059. 

Exercise  24.  —  Review. 
Find  the  value  of  : 

1.  1.4-1-2.08  +  3.895. 

2.  2.8  +  2.08-1-0.28+0.028  +  0.812. 

3.  1.667  +  0.4  +  0.286  +  6.08  +  0.636  +  0.931. 

4.  6.125  —  0.57. 

5.  (4.625 +  1.146) -(1.2 +  3.571). 

6.  6.913 -(2.85  —  0.937). 

7.  24  — 2.4 +  (5 -3.508)  — 3.092. 

8.  10  -  (4.25  -  2.5  +  2  —  0.625  —  0.4  —  2.02)  —0.295. 

9.  1.5  X  0.08  X  0.5. 

10.  0.1204X0.0168  X  100. 

11.  0.04X3.25X0.06. 

12.  36  X  0.002  X  2.05  X  0.00765. 

13.  0.139  X  28  +  42  X  0.002  +  6  X  0.004  —  0.05  X  20. 


52  DIVISION. 

14.    (10  - 1.25)  X  0.2  +  0.02  X  2.8  +  (80.3  X  0.1  -  5.3)  X 
10  —  805.3X0.02. 

15.  28.3696^-1.49.         19.    0.28744-^800. 

16.  0.27^-0.00225.  20.   491.205-^650. 

17.  8.8779 -f- 175.8.  21.   68.325-^6250. 

18.  0.0427^92.3.  22.    0.732 -f- 16,000. 

23.  1208.88 -f- 0.438. 

24.  2  -r  0.01  -  (0.2  -f-  0.02  +  0.8  *f- 10)  +  36.48  -^  8 

—  (4  -^-  0.05  —  2  +  0.6  -^  1.25). 

25.  72.2  -^- 10  -  2 -^  (0.5  -^  1.60)  +  2.125  -f-  (1.75  -  0.5). 

Exercise  25.  —  Review. 

1.  What  number  subtracted  88  times  from  80,005  will 
leave  13  as  a  remainder  ? 

2.  If  7  men  can  build  a  wall  in  16  days,  how  many  men 
will  it  take  to  build  a  wall  three  times  as  long  in  half  the 
time? 

3.  How  many  minutes  are  there  between  25  minutes 
past  8  in  the  morning  and  midnight? 

4.  If  the  velocity  of  sound  is  1090  feet  per  second, 
at  what  distance  is  a  gun  fired  the  report  of  which  I  hear 
11  seconds  after  seeing  the  flash?  (5280  feet  make  a 
mile.) 

5.  How  long  will  it  take  to  travel  30.2375  miles  at 
the  rate  of  8.85  miles  per  hour  ? 

6.  If  the  circumference  of  a  circle  is  3.1416  times  the 
diameter,  find  the  circumference  of  a  circle  whose  diameter 
is  6.8  feet ;  also,  find  the  diameter  of  a  circle  whose  cir- 
cumference is  20  inches. 

7.  How  much  wire  will  be  required  to  make  a  hoop  30 
inches  in  diameter,  allowing  2  inches  for  the  joining  ? 

8.  How  many  times  would  the  hoop  of  Ex.  7  turn 
in  going  half  a  mile  ? 


DIVISION.  53 

9.  Cork,  whose  weight  is  0.24  of  the  weight  of  water, 
weighs  15  pounds  per  cubic  foot.  What  is  the  weight  of 
6  cubic  feet  of  oak,  if  the  weight  of  oak  is  0.934  of  the 
weight  of  water  ? 

10.  From  what  number  can  847  be  subtracted  307  times 
and  leave  a  remainder  of  49  ? 

11.  What  is  the  235th  part  of  141,235  ? 

1 2.  What  will  343  barrels  of  flour  cost  at  $ 6.37  a  barrel  ? 

13.  Twelve  makes  a  dozen,  and  12  dozen  makes  a  gross. 
How  many  steel  pens  in  28  gross  ?  What  will  a  gross  of 
eggs  cost  at  27  cents  a  dozen  ? 

14.  How  much  must  be  added  to  $4429  to  make  the  sum 
equal  to  43  X  $241  ? 

15.  What  number  deducted  from  the  26th  part  of  2262 
will  leave  the  87th  part  of  the  same  number  ? 

16.  At  the  ordinary  rate,  123  words  a  minute,  how  long 
will  it  take  a  man  to  deliver  a  speech  of  15  pages,  if  each 
page  contains  28  lines,  and  each  line  11  words  ?  How  long 
would  it  have  taken  Daniel  Webster  to  deliver  the  same 
speech,  whose  rate  was^93  words  a  minute  ? 

17.  How  long  will  it  take  a  railway  train  to  go  from 
New  York  to  San  Francisco,  3310  miles,  at  the  rate  of 
1973  feet  a  minute  ? 

18.  How  many  hours  will  it  take  to  count  a  million,  at 
the  rate  of  67  a  minute  ? 

19.  If  you  put  into  a  box  17  cents  a  day,  including 
Sundays,  beginning  January  1  and  ending  July  4,  how 
much  money  will  there  be  in  the  box  ? 

20.  If  a  man's  income  is  $3000  a  year,  and  his  daily 
expenses  average  $7.68,  what   does    he    save  in  a  year  ? 

21.  In  a  question  of  division  the  quotient  was  87.83,  the 
divisor,  759.     What  was  the  dividend  ? 

22.  What  is  the  nearest  number  to  7196  that  will  con- 
tain 372  without  a  remainder  ? 


54  DIVISION. 

23.  It  is  3.1416  times  as  far  round  a  wheel  as  across  it. 
How  many  times  will  a  wheel  4.5  feet  across  turn  in 
going  23  miles  of  5280  feet  each  ? 

24.  How  many  gallons  of  231  cubic  inches  are  contained 
in  a  cubic  foot  of  1728  cubic  inches  ?  in  a  bushel  of  2150.42 
cubic  inches  ?  How  many  cubic  feet  in  a  bushel  ?  How 
many  bushels  in  31.5  gallons  ? 

25.  Seven  children  had  left  to  them  $7186  apiece  ;  one 
died,  and  his  share  was  divided  among  the  surviving  six. 
How  much  had  each  then  ? 

26.  How  long  will  it  take  2  men  to  do  what  1  man  can 
do  in  6  days  ?  what  4  men  can  do  in  3  days?  what  3  men 
can  do  in  4  days  ? 

27.  Divide  $1.80  among  Thomas,  Richard,  and  Henry  in 
such  a  way  that  Henry  shall  receive  3  cents  for  every 
5  cents  that  Thomas  gets,  and  Richard  shall  receive  2  cents 
for  every  3  cents  that  Henry  gets. 

28.  Divide  $87.84  between  B  and  C  so  that  C  shall  get 
$19  as  often  as  B  gets  $17. 

29.  Three  partners  received  for  goods:  one,  $371.63; 
the  second,  $285.40;  the  third,  $411.91.  They  paid  for 
the  goods  $879.34.  and  divided  the  profit  equally  among 
them.     How  much  did  each  receive  ? 

30.  If  there  are  12  inches  in  a  foot,  how  many  inches 
long  is  a  wall  35  feet  in  length  ?  If  a  brick  and  its  share 
of  mortar  is  8.4  inches  long,  how  many  bricks  in  length 
is  the  wall  ? 

31.  If  a  brick  and  its  mortar  is  2.4  inches  high,  how  many 
bricks  are  required  to  build  a  wall  12  feet  high,  35  feet 
long,  if  the  width  of  the  wall  is  the  width  of  two  bricks? 

32.  What  is  the  total  weight  of  the  wall  of  Ex.  31,  if  a 
brick  with  its  share  of  the  mortar  weighs  4.13  pounds  ? 
What  is  the  weight  after  a  long  rain,  when  the  weight  is 
increased  to  4.27  pounds  for  each  brick  ? 


DIVISION.  55 

33.  How  many  pounds  does  each  foot  in  length  of  the 
wall  of  Ex.  31  weigh  ? 

34.  If  60.98  cubic  inches  of  brick  weigh  4  pounds,  how 
many  cubic  inches  of  brick  weigh  1  pound  ?  How  many 
pounds  will  a  cubic  foot  (1728  cubic  inches)  weigh  ? 

35.  If  a  cubic  foot  of  water  weighs  62.5  pounds,  how 
many  times  as  heavy  as  water  is  brick  ? 

36.  Light  moves  through  the  air  at  the  rate  of  186,500 
miles  a  second.  How  many  times  can  it  go  around  the 
earth  in  a  second,  if  the  distance  round  the  earth  is 
24,897.714  miles? 

37.  Light  moves  through  the  air  at  the  rate  of  300,190 
kilometers  a  second.  How  many  times  can  it  go  around 
the  earth  in  a  second,  if  the  distance  round  the  earth  is 
40,007.5  kilometers? 

38.  A  minute  is  60  seconds.  How  many  miles  and  how 
many  kilometers  can  light  travel  through  air  in  a  minute  ? 

39.  An  hour  is  60  minutes.  How  many  miles  and  how 
many  kilometers  can  light  travel  in  an  hour  ? 

40.  The  distance  round  the  earth,  given  in  Ex.  37,  is 
measured  on  a  north  and  south  line.  Around  the  equator 
the  distance  is  40,075.45  kilometers.  How  many  times 
could  light  move  round  the  equator  in  one  minute  ? 

41.  Find  the  reciprocal  of  the  difference  between  31.24 
and  31.23768. 

42.  The  Hanoverian  mile  is  25,400  Hanoverian  feet 
long,  and  each  foot  is  0.9542  of  an  English  foot.  Find  to 
four  places  of  decimals  the  fraction  that  an  English  mile  of 
5280  English  feet  is  of  a  Hanoverian  mile. 

43.  Express  in  inches  the  length  of  a  meter,  given  that 
a  meter  is  one  ten-millionth  of  a  quarter  of  the  earth's  cir- 
cumference, that  the  circumference  is  3.14159  times  the 
diameter,  that  the  diameter  of  the  earth  is  7911.7  miles, 
and  that  a  mile  is  5280  X  12  inches, 


56 


DIVISION. 


44.  How  must  a  number  be  altered  that  its  reciprocal 
may  be  doubled  ? 

45.  What  effect  is  produced  on  the  sum  of  two  numbers, 
if  the  same  number  is  added  to  each  of  them  ?  What 
effect  on  the  difference  ? 

46.  What  effect  is  produced  on  the  product  of  two 
numbers,  if  both  numbers  are  multiplied  by  the  same 
number  ?     What  effect  on  the  quotient  ? 

47.  What  effect  is  produced  on  the  remainder,  if  both 
divisor  and  dividend  are  multiplied  by  the  same  number  ? 
If  both  are  divided  by  the  same  number  ? 

48.  In  going  from  one  planet  to  another,  light  probably 
moves  faster  than  in  air.  Suppose  it  moves  at  the  rate  of 
309,800  kilometers  a  second,  how  long  would  it  take  light 
to  perforin  each  of  the  following  journeys  : 


Moon  to  Earth 
Sun  to  Earth 
Sun  to  Mercury 
Sun  to  Venus 
Sun  to  Mars     . 
Sun  to  the  Asteroids 
Sun  to  Jupiter 
Sun  to  Saturn 
Sun  to  Uranus 
Sun  to  Neptune  . 
Sun  to  the  nearest  star 


375,600  kilometers. 
.     147,250,000 
.  56,900,000 
.     100,400,000  " 

224,100,000  " 

.     400,000,000 

765,400,000  " 

.  1,403,000,000 
.       2,817,000,000  " 

.  4,421,000,000 
24,000,000,000,000 


49.  A  kilometer  is  about  0.6214  of  a  mile.  How  many 
miles  is  each  of  the  planets  from  the  sun  ? 

50.  If  11.75  tons  of  coal  cost  $82.25,  what  will  21.4 
tons  cost  ? 

51.  Find  the  number  of  hours  it  will  take  a  locomotive 
running  at  the  rate  of  27  miles  an  hour  to  make  the  distance 
passed  over  in  13.25  hours  by  another  locomotive  that  has 
a  velocity  of  43.5  miles  an  hour. 


DIVISION.  57 


Review  Questions. 

What  are  units  ?  numbers  ?  integral  numbers  ?  decimal  numbers  ? 
abstract  numbers  ?   concrete  numbers  ?   like  numbers  ? 

What  is  notation  ?  numeration  ?  Where  is  the  decimal  point 
placed  ?  What  do  figures  in  the  first  place  to  the  left  of  the  decimal 
point  represent  ?  in  the  second  place  ?  in  the  third  place  ?  in  the 
fourth  place  ?  in  the  first  place  to  the  right  of  the  decimal  point  ?  in 
the  second  place  ?  in  the  third  place  ?  in  the  fourth  place  ?  Which 
place  do  the  units  of  a  number  occupy  ?  the  tens  ?  the  hundreds  ? 
the  thousands  ?  the  tenths  ?  the  hundredths  ?  the  thousandths  ?  At 
what  rate  does  the  value  of  a  figure  increase  from  right  to  left  ?  In 
separating  a  row  of  figures  into  periods  where  do  we  begin  ?  How 
many  figures  in  each  period  ?  What  is  the  period  on  the  right  called  ? 
the  second  period  ?  the  third  period  ?  Which  period  do  we  write 
first  ?  Which  period  do  we  read  first  ?  How  do  we  write  a  decimal  ? 
How  do  we  read  a  decimal  ?  In  reading,  by  what  word  do  we  connect 
the  integral  and  decimal  parts  of  a  number  ? 

What  is  addition?  What  is  the  result  called?  What  kind  of 
numbers  only  can  be  added  ?  What  is  the  sign  of  addition  ?  Does 
it  make  any  difference  in  what  order  the  numbers  are  added  ?  In 
addition  how  do  we  arrange  the  decimal  points  ?  How  do  we  prove 
that  the  work  of  addition  Ts  correct  ? 

What  is  subtraction  ?  What  is  the  greater  number  called  ?  the 
smaller  ?  the  result  ?  What  kind  of  numbers  must  the  minuend, 
subtrahend,  and  remainder  be  ?  What  is  the  sign  of  subtraction  ? 
In  subtraction,  how  do  we  arrange  the  decimal  points  ?  How  do  we 
prove  that  the  work  of  subtraction  is  correct  ? 

What  is  multiplication  ?  the  multiplicand  ?  the  multiplier  ?  the 
product  ?  What  kind  of  a  number  must  the  multiplier  be  ?  What 
are  the  factors  of  a  product  ?  Does  it  make  any  difference  in  what 
order  the  factors  are  multiplied  ?  What  is  the  sign  of  multiplication  ? 
How  many  decimal  places  must  the  product  have  ?  How  do  we  prove 
that  the  work  of  multiplication  is  correct  ? 

What  is  division  ?  the  dividend  ?  the  divisor  ?  the  quotient  ?  If 
the  divisor  is  not  an  integer  how  can  we  make  it  an  integer  without 
altering  the  quotient  ?  If  the  divisor  is  an  integer  where  do  we  place 
the  decimal  point  in  the  quotient  ?  What  is  the  advantage  of  writing 
the  quotient  over  the  dividend  in  long  division  ?  What  is  the  sign  of 
division  ?     How  do  we  prove  that  the  work  of  division  is  correct  ? 


CHAPTER   V. 

METRIC   MEASURES. 

110.  To  measure  a  quantity  is  to  find  the  number  of 
times  it  contains  a  known  quantity  of  the  same  kind,  called 
the  unit  of  measure. 

111.  The  metric  system  is  a  system  of  weights  and 
measures  expressed  in  the  decimal  scale. 

112.  The  standard  meter,  as  defined  by  law,  is  the 
length  of  a  bar  of  very  hard  metal,  carefully  preserved  at 
Paris,  accurate  copies  of  which  are  furnished  the  govern- 
ments of  all  civilized  countries. 

The  meter  was  intended  to  be  one  ten-millionth  of  the  distance 
from  the  equator  to  the  north  pole,  but  more  careful  measurements 
show  that  this  distance  is  10,001,887  meters. 

113.  The  principal  units  of  the  metric  system  are  : 

The  meter  (m)  for  lengths ; 
The  square  meter  (<im)  for  surfaces ; 
The  cubic  meter  (ebm)  for  large  volumes; 
The  liter  Q)  (lee'ter)  for  smaller  volumes  j 
The  gram  (g)  for  weights. 

114.  All  these  units  are  divided  and  multiplied  deci- 
mally, and  the  size  of  the  measures  thus  produced  is  shown 
by  a  prefix  ;  namely,  deka,  meaning  10  ;  hekto,  meaning 
100  ;  kilo,  meaning  1000  ;  myria,  meaning  10,000 ;  and  deci, 
meaning  0.1  ;  centi,  meaning  0.01 ;  milli,  meaning  0.001. 

As  in  United  States  money  we  seldom- speak  of  any- 
thing except  dollars  and  cents,  so  in  metric  measures 
only  measures  printed  in  black  letter  are  in  common  use. 


METRIC   MEASURES.  59 

Measures  of  Length. 

115.  The  principal  unit  of  length  is  the  meter. 

Table. 

10  millimeters  (mm)  =  1  centimeter  (om). 

10  centimeters  ss  1  decimeter  (dm). 

10  decimeters  =  1  meter  (m). 

10  meters  =  1  dekameter  (dkm). 

10  dekameters  =  1  hektometer  (hm). 

10  hektometers  =  1  kilometer  (km). 

10  kilometers  =  1  myriameter. 

Note.  The  names  of  all  compound  units  are  accented  H 
on  the  first  syllable  ;  thus,  millimeter,  kilometer.  <? 

116.  A  length  given  in  one  unit  may  be  ex-  g- 
pressed  in  another  unit  by  simply  moving  the  ■ 
decimal  point.  o 


Thus,  17,856,342mm  may  be  written  as  kilo-meters  by      § 
observing  that  milli-meters  are  changed  to  meters  by      2 
moving  the  point  three  places  to  the  left;  and  these 
meters   into    kilo-meters  Joy    carrying    it    three   places      K 
further,  making  in  all  six  places.      Therefore, 
17,856,342""!!=:  i7.856342km. 

Again,  4.876326km  may  be  written  as  centi-meters  by 
observing  that  kilo-meters  are  changed  to  meters  by 
moving  the  point  three  places  to  the  right,  and  meters 
to  centi-meters  by  moving  it  two  places  further,  making 
in  all  Jive  places.     Therefore, 

4.876326km  =  487,632.6cm 


r, 


g 


117.  The  rule,  therefore,  for  this  conversion  is  : 

First  count  the  number  of  places  needed  to  convert  the  given 
measure  into  terms  of  the  principal  measure;  then  the  number 
needed  to  convert  the  principal  into  the  required  measure. 

118.  Before  adding  or  subtracting,  the  quantities  must 
be  written  in  terms  of  the  same  unit  of  measure. 


60  METRIC   MEASURES. 

Exercise  26. 

1.  Change   5427m  to  kilometers  ;    to   millimeters  ;   to 
centimeters. 

2.  How  many  meters   in    6853mm  ?   how  many  centi- 
meters ?  what  part  of  a  kilometer  ? 

3.  Write  49.7m  as  centimeters  ;  as  millimeters  ;  as  the 
decimal  of  a  kilometer. 

4.  How  many  centimeters  in  12.4km  ?  how  many  milli- 
meters ? 

5.  Change  1230m  to  kilometers  ;  to  centimeters. 

6.  Write  1230cm  as  meters  ;  as  millimeters. 
Find  in  meters  the  value  of  the  following  : 

7.  0.435m  +  852cm  +  4263mm  +  O.^S1"" 

8.  0.927km  —  6495cm  ;  4.37cra  —  42.87mm. 

9.  8  X  0.0457km  ;  3.04  X  60.93cm;  5.43  X  67.2mm. 

10.  38,01 9mm  4- 0.097;  0.41km-h  25.625. 

11.  At  fl.87  a  meter,  what  is  the  cost  of  6.20m  of 
cloth  ? 

12.  At  $0.75  a  meter,  what  is  the  cost  of  60™  of  cloth  ? 

13.  From  a  piece  of  cloth  containing  47.60m  a  tailor 
cuts  off  three  pieces  :  the  first  of  3.80m,  the  second  of 
1.30m,  and  the  third  of  45cm.  How  many  meters  of  the 
cloth  are  left  ? 

14.  What  is  the  value  of  60cm  of  cloth  at  $5.20  a  meter  ? 

15.  If  $6.00  is  paid  for  a  railroad  ticket  to  travel 
440km,  what  is  the  fare  per  kilometer  ? 

16.  If  a  train  goes  288km  in  9  hours,  how  many  meters 
does  it  go  in  a  minute  ?     (1  hour  =  60  minutes.) 

17.  If  a  man  walks  at  the  rate  of  6km  an  hour,  what 
part  of  an  hour  will  it  take  hini  to  walk  420m  ? 

18.  A  railroad  carried  412  passengers  18km  for  $88,992  ; 
at  the  same  rate,  what  will  it  receive  for  carrying  350 
passengers  35km  ? 


METRIC    MEASURES.  61 


Measures  of  Surface. 

119.  A  square  is  a  flat  surface  with  four 
square  corners  and  four  equal  straight  sides. 

120.  A  square  meter  is  a  square  one  meter 

Square.  on   a  side> 

The  principal  unit  of  surface  is  the  square  meter. 

121.  Suppose  the  square  in  the  mar- 
gin to  represent  a  square  meter.  It  is 
divided  into  ten  equal  horizontal  bands, 
and  each  band  is  one  tenth  of  the  square 
meter.  Each  band  can  be  divided,  as 
the  upper  one  is,  into  ten  little  squares 


i  i  i  i  i  i  i  i  i: 


measuring  one  tenth  of  a  meter  on  a  side.  Each  of  these 
squares  will  be  0.1  of  the  band,  or  0.01  of  the  whole 
square.  The  square  meter,  therefore,  contains  10  X  10,  or 
100  square  decimeters. 

In  like  manner,  a  square  decimeter  contains  100  square 
centimeters,  and  therefore  a  square  meter  contains  100  X 
100,  or  10,000  square  centimeters. 

In  like  manner,  a  square  meter  contains  1000  X  1000,  or 
1,000,000  square  millimeters. 

Table. 

100  square  millimeters  (imm)  =  1  square  centimeter  (<icm). 

100  square  centimeters  =  1  square  decimeter  (<idm). 

100  square  decimeters  =  1  square  meter  (im). 

100  square  meters  =  1  square  dekameter  (idkm). 

100  square  dekameters  =  1  square  hektometer  (ihm). 

100  square  hektometers  =  1  square  kilometer  (<ikm). 

122.  In  measures  of  length  each  unit  is  io  times  as 
large  as  the  next  smaller  unit,  but  in  measures  of  surface 
each  unit  is  ioo  times  as  large  as  the  next  smaller  unit. 


62  METRIC    MEASURES. 

123.  In  the  measurement  of  land,  the  square  meter  is 
called  a  centar,  the  square  dekameter  is  called  an  ar,  and 
the  square  hektometer  is  called  a  hektar. 

Table  of  Land  Measures. 

100  centers  (ca)       =  1  ar  (•). 
100  ars  =  1  hektar  (ha). 

124.  In  surface  measures,  when  the  ar  is  the  unit,  ioo 
ars  make  a  hektar  ;  but  when  the  square  meter  is  the  unit, 
ioo  X  ioo,  or  10,000  square  meters  make  a  square  hekto- 
meter or  hektar. 

Exercise  27. 

1.  Change  l,854,276qm  to  hektars  ;  to  square  kilome- 
ters. 

2.  How  many  hektars  in  2.7856qkm  ? 

3.  Write    1.7431qm   as  square  centimeters ;  as  square 
millimeters. 

4.  How  many  square  kilometers  in  17,467.5h*  ? 

5.  How  many  square  meters  in  1.3614qkm  ? 

6.  How  many  square  meters  in  2.25ha? 

7.  How  many  square  centimeters  in  0.0137qm  ? 

8.  Write  3.57 lqcm  as  square  millimeters. 

9.  If  a  field  contains    7500ca,  how  many  ars  does  it 
contain  ?     What  part  of  a  hektar  ? 

10.  How  many  square  meters  must  be  added  to  22,612qm 
to  make  4ha  62a  17ca  ? 

11.  A  field  containing  72.4a  is  sold  at  15  cents  a  square 
meter.     What  is  received  for  the  field  ? 

12.  If  C2a  12°*  of  land  is  sold  for  $1366.64,  what  is  the 
price  per  square  meter? 

13.  How  many  square  centimeters  must  be  taken  from 
12,473qcm  to  leave  lqm  14qdm  53qcm  ? 


METRIC    MEASURES. 


63 


Measures  of  Volume. 

125.  A  cube  is  a  solid  bounded  by  six  equal  squares. 
Each  bounding  square  is  called  a  face,  and 
the  intersection  of  two  faces  is  called  an 
edge. 


126.  A  cubic  meter  is  a  cube  one  meter  on 
Cube-         an  edge. 
The  principal  unit  of  volume  is  the  cubic  meter. 

127.  The  cubic  meter  can  be  divided  into  10  layers, 
each  a  meter  square  and  a  decimeter 
thick.  Each  layer  will,  therefore,  be 
0.1  of  a  cubic  meter.  Again,  each 
layer  can  be  divided  into  10  equal 
parts.  Each  part  will,  therefore,  be 
0.1  of  the  layer,  or  0.01  of  the  meter, 
and  will  be  a  decimeter  square  and 
a  meter  long.  Also,  each  one  of 
these  parts  can  be  divided  into  10 
equal  parts,  each  of  which  will  be  a  cubic  decimeter,  and 
will  be  0.1  of  0.01,  or  0.001  of  the  cubic  meter. 

The  cubic  meter,  therefore,  contains  1000  cubic  decime- 
ters. 

In  like  manner,  a  cubic  decimeter  contains  1000  cubic 
centimeters,  and  a  cubic  centimeter  contains  1000  cubic 
millimeters. 


/ 

A 

- 

p  fi 

'///////  / ../. 

Table. 


1 1000  cubic  millimeters  (cmm) 
1000  cubic  centimeters 
1000  cubic  decimeters 


=  1  cubic  centimeter  (ccm). 
=  1  cubic  decimeter  (cdm). 
=  1  cubic  meter  (cbm). 


128.   In  measures  of  volume  each  unit  of  measure  is 
iooo  times  as  large  as  the  next  smaller  unit  of  measure. 


64  METRIC   MEASURES. 

129.   In  the  measurement  of  wood,  the  cubic  meter  is 
called  a  ster. 


Ster  of  Wood. 

Table  of  Wood  Measures. 

10  decisters  (d*1)  =  1  ster  (8t). 

10  sters  =  1  dekaster  (dk8t). 

Exercise  28. 

1.  Write  2.25cbm  as  cubic  centimeters. 

2.  Change  2,162,875ccm  to  cubic  meters. 

3.  Change  0.0175cbm  to  cubic  millimeters. 

4.  Change  46,164ccm  to  cubic  decimeters. 

5.  What  is  the  equivalent  of  0.875dk8t  in  cubic  meters  ? 
in  cubic  centimeters  ? 

6.  How  many  sters  are  there  in   14.75dk8t  of  wood? 
how  many  decisters  ? 

7.  What  is  the  cost  of  28.25dk8t  of  wood  at  $1.25  a 
ster  ? 

8.  Find  the  cost  of  an  oak  beam  containing  1250cdm  at 
$25  a  cubic  meter. 

9.  How   many   cubic   centimeters   must   be  added  to 
l,262,376ccm  to  make  2cbm  2cdm  2ccm? 

10.   How  many  cubic  millimeters  must  be  taken  from 
22,350,000,000cmm  to  leave  20cbm  22cdm  222ccm  ? 


METRIC    MEASURES. 


65 


Measures  of  Capacity. 

130.  The  principal  unit  of  capacity  is  the  liter. 

131.  In  measuring  liquids,  grain,  etc.,  the  cubic  deci- 
meter is  called  a  liter. 


Table. 


10  milliliters  (ml) 
10  centiliters 
10  deciliters 
10  liters 
10  dekaliters 
10  hektoliters 


=  1  centiliter  (cl). 
=  1  deciliter  (<U). 
=  1  liter  (*). 
=  1  dekaliter  (*"). 
=  1  hektoliter  (hl). 
=  1  kiloliter  (*i). 


132.   In  measures  of  capacity  each  unit  of  measure  is 
io  times  as  large  as  the  next  smaller  unit  of  measure. 


Exercise  29. 

1.  How  many  liters  in  1.7cbm?  in  157,854ccm  ? 

2.  How  many  cubic  centimeters  in  9.51?  in  0.0151? 

3.  Change  1.25w  to  cubic  centimeters ;  to  the  fraction 
of  a  cubic  meter. 

4.  Change  431.881  to  hekto- 
liters; to  the  fraction  of  a  cubic 
meter. 

5.  Write  0.375cbm  as  liters;  as 
cubic  centimeters. 

6.  Write  734,159.651ccm  as 
liters ;  as  hektoliters  ;  as  cubic 
meters. 

7.  How  many  cubic  meters  in  8,573,412.867CCIn  ? 

8.  Change  0.734578912cbm  to  cubic  centimeters ;  to  liters. 

9.  Change   1731. 51  to   cubic  meters  ;   to   cubic   centi- 
meters. 


Liter  =  Cubic  Decimeter. 


66  METRIC   MEASURES. 


Measures  of  Weight. 

133.  The  units  of  weight  are  the  weights  of  units  of 

pure  water  taken   at   its   greatest 

0  density,  that  is,  a  little  above  the 

freezing  point. 
cA^  The  principal  unit  is  the  gram, 

which  is  the  weight  of  a  cubic  cen- 

Cubic  Centimeter.    Gram  Weight,     timeter  of  Water. 

Table. 

10  milligrams  (m«)        =  1  centigram  (<*). 

10  centigrams  =  1  decigram  (d«). 

10  decigrams  =  1  gram  (*). 

10  grams  =  1  dekagram  (dk«). 

10  dekagrams  =  1  hektogram  (*>*). 

10  hektograms  =  1  kilogram  (k«). 

1000  kilograms  =  1  metric  ton  (*)• 

Note.     The  kilogram  is  often  called  a  kilo. 

134.  In  measures  of  weight  each  unit  of  weight  is  io 
times  as  great  as  the  next  smaller  unit  of  weight. 

135.  A  cubic  centimeter  of  water  weighs  a  gram. 
A  liter  of  water  weighs  a  kilogram. 

A  cubic  meter  of  water  weighs  a  ton. 


Exercise  30. 

1.  How  many  kilos  in  1.73'?  in  0.341  of  a  ton  ? 

2.  How  many  kilos  will  a  hektoliter  of  water  weigh  ? 

3.  Change  13,756mg  to  grams  ;  to  the  fraction  of  a  kilo. 

4.  What  is  the  weight  in  grams  of  346.1ccm  of  water  ? 

5.  Find  the  weight  in  kilograms  of  0.37615cbm  of  water. 

6.  Change  0.6778kg  to  milligrams. 

7.  How  many  milligrams  in  the  third  part  of  17.4*  ? 


METRIC   MEASURES.  67 


Exercise  31. 


1.  Add  17.3m,  87.41m,  271cm,  380mm,  and  1.79m. 

2.  Add  15.87m,  394.6dm,  47.52ra,  7538cm,  and  75.89m. 

3.  Add  187cm,  49.3ra,  317mm,  and  6.138m. 

4.  In  a  room  the  doorsill  is  3cm  high;  the  door,  2.34m; 
the  finish  over  the  door,  13.7cm ;  and  the  distance  from  the 
finish  to  the  ceiling  is  93cm.  What  is  the  height  of  the 
room  ? 

5.  The  distance  to  the  post  office  is  3.31km  ;  thence  to 
the  mill,  1.711km;  thence  to  the  store,  3.718km  ;  thence 
home,  2.543km.     How  long  is  the  circuit  ? 

6.  The  distance  from  Portland,  Me.,  to  Boston  is  174km  ; 
Boston  to  Albany,  317km  ;  Albany  to  Buffalo,  478kra  ;  Buffalo 
to  Chicago,  863km ;  Chicago  to  Omaha,  789km;  Omaha  to 
Cheyenne,  830km.  How  far  is  it  from  Cheyenne  to  Port- 
land ?  from  Cheyenne  to  Albany  ?  from  Boston  to  Chicago  ? 
from  Boston  to  Cheyenne? 

7.  If  I  travel  789.7km  a  day,  how  far  shall  I  go  in  7 
days  ?  in  8.5  ?  in  19.6?  in  27.8  ?  in  365  ? 

8.  How  much  will  3m  of  cloth  cost  at  $1.37  a  meter  ? 
How  much  will  5.38m  cost  at  $2.63 -a  meter  ? 

9.  How  much  will  13.4ks  of  opium  be  worth  at  $8.48 
a  kilo  ?  28.79k*  at  $7.96  a  kilo  ? 

10.  If   one  barrel   of   flour  weighs  88.9kg,   how  many 
barrels  can  be  filled  from  444.5*  of  flour  ? 

1 1 .  How  many  steps  80cm  long  will  a  man  take  in  walk- 
ing a  kilometer  ? 

12.  At  16   cents  a  liter,  what  is  the  cost  of  52.4W  of 
olive  oil  ? 

13.  What  is  the  cost  of  6dk8t  48t  of  oak  wood  at  $1.75 
per  ster  ? 

14.  If  a  pasture  contains  22,408ca,  how  many  ars  does 
it  contain  ?  how  many  hektars  ? 


68  METRIC   MEASURES. 

136.  A  flat  surface  bounded  by  straight  lines  or  by  a 
curved  line  is  called  a  plane  figure. 

137.  A  circle  is  a  plane  figure 
bounded  by  a  curved  line  called  the 
circumfere7ice,  all  points  of  which 
are  equally  distant  from  a  point 
within  called  the  centre. 

A  straight  line  drawn  through 
the  centre,  having  its  ends  in  the 
circumference,  is  called  a  diameter  ; 
circle-  and  half  a  diameter  is  called  a  radius. 

138.  If  the  diameter  of  a  circle  is  multiplied  by  8.1416, 
the  product  is  the  length  of  the  circumference. 

15.  Find  the  circumference  of  a  circle  lm  in  diameter. 

16.  Find  to  the  nearest  tenth  of  a  millimeter  the  circum- 
ferences of  circles  whose  diameters  are  respectively  83m; 
3.71m;  32.8m;  10.4cm  ;  11.8cm  ;  167.1mm  ;  39.3mm. 

17.  What  is  the  length  of  the  earth's  orbit,  to  the 
nearest  meter,  if  the  diameter  of  the  orbit  is  294,481,21 7km  ? 

18.  What  is  the  circumference  of  a  carriage  wheel,  1.31m 
in  diameter  ?   How  far  will  it  go  in  turning  once  ?  17  times  ? 

19.  How  many  times  must  the  wheel  of  Ex.  18  turn  in 
going  69.429m  ?  73.513ra?  17.27km? 

20.  Find  the  reciprocal  of  3.1416  to  the  fifth  place. 

139.  From  Ex.  20,  if  a  circumference  is  multiplied  by 
0.81881,  the  product  is  the  diameter. 

21.  How  thick  through  is  a  tree  whose  girth  is  2.97m  ? 

22.  What  is  the  diameter  of  a  wheel  that  turns  19.5 
times  in  going  107. 25m  ? 

23.  What  is  the  diameter  of  a  rope  of  which  the  cir- 
cumference is  20cm  ? 


METRIC    MEASURES. 


Areas. 

140.  A  surface  has  two  dimensions,  length  and  breadth. 

141.  The  unit  of  surface  is  a  square,  each  side  of  which 
is  a  unit  of  length. 

142.  The  area  of  a  surface  is  the  number  of  square 
units  it  contains. 

143.  The  perimeter  of  a  plane  figure  is  the  distance 
round  it. 

144.  A  rectangle  is  a  plane  figure  with  four  straight 
sides  and  four  square  corners. 

145.  Suppose  the  rectangle  in  the  margin  is  3 
and  2cm  wide.  If  lines  are  drawn  as 
represented,  the  surface  is  divided 
into  square  centimeters.  There  are 
2  horizontal  rows  of  3qcm  each ;  that 
is,  in  all,  2  X  3<*cm.     Hence, 


long 


To  Find  the  Area  of  a  Rectangle, 

Express  the  length  and  breadth  of  the  rectangle  in  the  same 
linear  unit ;  take  the  product  of  these  two  numbers  for  its 
area  in  square  units  of  the  same  name  as  the  linear  unit. 

The  number  of  square  units  in  a  rectangle  divided  by  the 
number  of  linear  units  in  one  dimension  gives  the  number  of 
linear  units  in  the  other  dimension. 

Exercise  32. 

1.  Find  the  area  of  a  rectangle  17cm  by  19cm. 

2.  In  a  rectangular  township  16km  by  7km,  how  many 
hektars  ?  If  there  are  in  it  47.3km  of  highway,  averaging 
11. 7m  wide,  how  much  land  is  left  for  other  uses  ? 

3.  In  a  rectangular  field  751.3m  long  and  189.3m  wide 
is  a  rectangular  garden  31.4m  by  17.8m.  How  many  hektars 
in  the  field  ?     How  many,  exclusive  of  the  garden  ? 


70  METRIC   MEASURES. 

4.  If  my  garden  contains  941.65qm  and  my  neighbor's 
747.37qm,  what  is  the  area  in  hektars  of  both  taken  to- 
gether ? 

5.  If  a  painter  can  cover  8.786qm  in  an  hour,  how  many 
square  meters  can  he  cover  in  1.78  hours  ?  in  3.86  hours  ? 
in  4.57  hours  ? 

6.  How  many  hektars  in  each  of  three  rectangular 
fields  :  one  measuring  315.71m  by  78.91m  ;  a  second,  293.6m 
by  84.84m  ;  the  third,  346.8m  by  71.82m  ?  How  many  in 
the  three  together  ? 

7.  Find  the  price  of  a  rectangular  field,  346.8m  by  71.82m, 
at  $67.50  a  hektar ;  at  $384  a  hektar  ;  and  at  $2,375  a 
square  meter. 

8.  Find  the  length  of  a  rectangle  17cm  wide  that  con- 
tains 306qcm.  What  length  of  carpet  75cm  wide  is  required 
to  make  27qm? 

9.  A  room  is  16m  long,  8ra  wide,  and  8m  high  ;  another 
room  is  7m  long,  7m  wide,  and  3m  high.  How  many  square 
meters  of  painting  on  the  walls  of  both  rooms,  if  no  allow- 
ance is  made  for  doors  and  windows  ?  How  many  more 
square  meters  of  painting  on  the  walls  of  the  larger  room 
than  on  those  of  the  smaller  ? 

146.   To  Find  the  Area  of  a  Circle, 

Multiply  the  square  of  the  radius  by  3.1416  ;  or,  multiply 
the  square  of  the  diameter  by  0.7854  (i  of  8.1 416). 

10.  -What  is  the  area  of  a  circle  27cm  in  diameter?  of 
a  circle  lm  in  diameter  ? 

1 1 .  What  is  the  area  in  hektars  of  a  circular  field  784m 
in  diameter  ? 

12.  Find  the  area  of  a  circle  31cm  in  diameter. 

13.  Find  the  area  of  a  circle  whose  radius  is  24m. 

14.  If  a  circle  has  a  radius  of  7cm,  how  many  square 
centimeters  does  it  contaiu  ? 


METRIC   MEASURES.  71 

15.  In  a  rectangular  sheet  of  zinc  1.76ra  long  and  89cm 
wide  are  two  circular  openings,  one  of  which  has  a  radius 
of  10.5crn,  the  other  a  radius  of  9.2cm.  What  is  the  area  of 
the  zinc  left  ? 

16.  A  piece  of  land  in  the  form  of  a  circle  has  a  radius 
of  40m  ;  in  the  middle  of  it  is  a  pond  forming  a  circle  of 
15m  radius.  What  is  the  total  surface  ?  the  surface  of  the 
pond  ?  the  surface  of  the  land  to  cultivate  ? 

17.  How  deep  is  a  well,  if  the  wheel  whose  diameter  is 
75cm  makes  26  revolutions  in  raising  the  bucket  ? 

147.  A  sphere  is  a  solid  bounded  by  a  curved  surface, 
all  points  of  which  are  equally  distant  from  a  point  within 
called  the  centre. 

148.  A  straight  line  drawn  through  the  centre  of  a 
sphere,  having  its  ends  in  the  surface,  is  called  a  diameter  ; 
half  a  diameter  is  called  a  radius. 

149.  The  area  of  the  surface  of  a  sphere  is  four  times 
the  area  of  a  circle  of  the  same  diameter.     Hence, 

To  Find  the  Area  of  the  Surface  of  a  Sphere, 

Multiply  the  square  of  the  diameter  by  3.1^16. 

18.  How  many  square  centimeters  of  surface  on  a  ball 
7cm  in  diameter  ? 

19.  How  many  square  centimeters  of  surface  on  a  ball 
18cm  in  diameter  ? 

20.  How  many  square  meters  of  surface  on  a  hemis- 
pherical dome  11.27m  in  diameter  ? 

Note.     A  hemisphere  is  half  a  sphere. 

21.  What  is  the  interior  surface  of  a  hemispherical 
basin  12cm  in  diameter  ? 

22.  What  is  the  interior  surface  of  a  hemispherical 
vase  70cm  in  diameter  ? 


72  METRIC    MEASURES. 


Carpeting  Rooms. 

150.  Carpeting  is  made  of  various  widths  and  is  sold 
by  the  length. 

In  determining  the  number  of  meters  required  for  a 
room,  we  first  decide  whether  the  strips  shall  run  length- 
wise or  across  the  room,  and  then  find  the  number  of  strips 
needed.  The  number  of  meters  in  a  strip,  including  the 
waste  in  matching  the  pattern,  multiplied  by  the  number 
of  strips  will  give  the  required  number  of  meters. 

23.  How  many  meters  of  carpet  60cm  wide  will  be  re- 
quired for  a  room  6m  long  and  5.4m  wide,  the  strips  running 
lengthwise  ?  how  many  meters  would  be  required  if  the 
carpet  were  80cm  wide  ? 

Since  the  room  is  640cm  wide,  it  will  take  ^-,  or  9  widths  of  car- 
pet CO01"  wide  ;  that  is,  9  X  6m,  or  54m,  will  be  required.  If  the  car- 
pet were  80cm  wide,  it  would  take  ^8^,  or  7  widths.  Six  widths  would 
leave  a  surface  ({O0™  wide  to  be  covered.  This  surface  would  require 
another  strip,  of  which  a  width  of  20cm  would  be  turned  under. 

24.  How  many  meters  of  carpet  56cm  wide  will  be  re- 
quired for  a  room  8.32m  long  and  6.6m  wide,  strips  running 
lengthwise  ? 

25.  How  many  meters  of  carpet  70cm  wide  will  be  re- 
quired for  a  room  7m  long  and  5.4m  wide,  strips  running 
across  the  room  ? 

26.  How  many  meters  of  carpet  80cm  wide  will  be  re- 
quired for  a  room  6m  long  and  5.47m  wide,  strips  running 
across  the  room  ? 

27.  How  many  meters  of  carpet  90cm  wide  will  be  re- 
quired for  a  room  5m  long  and  4.5m  wide,  strips  running 
lengthwise  ?     How  much  will  it  cost,  at  $1,875  a  meter  ? 

28.  How  many  meters  of  carpet  75cm  wide  will  be  re- 
quired for  a  room  5.25™  long  and  4.75m  wide,  strips  running 
across  the  room?     Find  the  cost,  at  $2,125  a  meter. 


METRIC    MEASURES.  73 

29.  How  many  meters  of  carpet  75cm  wide  will  be  re- 
quired for  a  room  5.6m  square  ?  How  wide  a  strip  will 
have  to  be  turned  under  ?  How  much  will  the  carpet  cost, 
at  $1.25  a  meter  ? 

Papering  and  Plastering. 

151.  The  area  of  the  four  walls  of  a  room  is  equal  to 
that  of  a  rectangle  whose  length  is  the  perimeter  of  the 
room,  and  whose  breadth  is  the  height  of  the  room. 

Perimeter  =  twice  the  length  -4-  twice  the  breadth. 
Area  =  height  X  perimeter. 

30.  Find  the  area  of  the  walls  of  a  room  whose  length 
is  6.12m,  breadth  5.05m,  and  height  3.5m. 

Perimeter  =  2  X  (6.12m  +  5.05m)=  22.34m. 
Area  =  (3.5  X  22.34)  v*=-  78.19<im. 

31.  How  many  rolls  of  paper  45cm  wide  and  8m  long, 
allowing  11. l^™1  for  doors  and  windows,  will  be  required 
to  paper  the  room  of^Ex.  30  ? 

32.  Find  the  cost  of  papering  a  room  8m  long,  5.5m  wide, 
and  4.5m  high,  with  paper  50cm  wide  and  7.5m  in  a  roll,  at 
$1.25  a  roll,  put  on;  if  there  is  a  baseboard  25cm  wide 
running  round  the  room,  and  an  allowance  of  llqm  is  made 
for  doors  and  windows. 

33.  Find  the  cost  of  plastering  the  room  of  Ex.  32,  at 
$0.50  a  square  meter. 

34.  Find  the  cost  of  papering  a  room  5.5m  long,  4.8m 
wide,  and  3.2m  high,  with  paper  45cm  wide,  7.5m  in  a  roll, 
at  $0,875  a  roll,  put  on,  allowing  12<im  for  baseboard, 
doors,  etc. 

35.  Find  the  cost  of  plastering  the  room  of  Ex.  34,  at 
$0.45  a  square  meter. 

36.  Find  the  cost  of  papering  a  room  6m  square  and 
3.5m  high,  with  paper  45cm  wide  and  7.5m  in  a  roll,  at 


74  METRIC    MEASURES. 

$0.75  a  roll,  put  on  ;  and  of  putting  on  a  border,  at  5 
cents  per  running  meter. 

37.  Find  the  cost  of  plastering  the  room  of  Ex.  36,  at 
$0.36  a  square  meter. 

38.  Find  the  cost  of  papering  a  room  13™  long,  12m 
wide,  and  7m  high,  with  paper  45cm  wide  and  7.5m  in  a  roll, 
at  $1.50  a  roll,  put  on ;  and  of  putting  on  a  border,  at 
$0.30  a  running  meter,  allowing  115qm  for  baseboard, 
doors,  etc. 

39.  Find  the  cost  of  plastering  the  room  of  Ex.  36,  at 
$0.60  a  square  meter. 

Board  Measure. 

152.  Boards  25m,n  or  less  in  thickness  are  sold  by  the 
square  meter. 

Boards  more  than  25mm  in  thickness  and  squared  lumber 
are  sold  by  the  number  of  square  meters  of  boards  25mm  in 
thickness  to  which  they  are  equivalent. 

Thus,  a  board  4m  long,  25cm  wide,  and  25mm  thick  contains  lm 
board  measure;  if  less  than  25mm  thick  it  still  contains  lm  ;  but  if 
76mm  thick,  it  contains  3m  board  measure,  for  it  is  equivalent  to 
three  boards  4m  long,  25cm  wide,  and  25mm  thick.     Hence, 

153.  To  Find  the  Board  Measure  of  Boards  more  than 
2gmm  thick  and  of  Squared  Lumber, 

Express  the  length  and  width  in  meters,  and  the  thickness 
in  millimeters  ;  divide  the  product  of  these  three  numbers  by 
25  for  the  number  of  meters  board  measure. 

If  a  board  tapers  regularly,  its  average  width  is  found 
by  taking  one-half  the  sum  of  its  end  widths. 

How  many  meters,  board  measure  : 

40.  In  a  board  8m  long,  20cm  wide,  and  20mm  thick  ? 

41.  In  a  joist  5m  long,  25cm  wide,  and  75mm  thick  ? 

42.  In  a  stick  of  timber  15m  long  and  40cm  square  ? 

43.  In  2  joists  5m  long,  27.5cm  wide,  and  50mm  thick  ? 


METRIC    MEASURES.  75 

44.  How  many  meters,  board  measure,  in  10  planks, 
each  4m  long,  45cm  wide,  and  10cm  thick?  What  is  the 
value  of  these  planks,  at  $25  a  hundred  meters  ? 

45.  How  many  meters,  board  measure,  in  25  box  boards, 
each  4ra  long,  42cm  wide,  and  20mm  thick  ?  What  is  their 
value,  at  $14  a  hundred  meters  ? 

Find  the  cost  of  : 

46.  Ten  joists  4.5m  long,  10cm  wide,  and  7.5cm  thick,  at 
$11  a  hundred  meters. 

47.  Thirty-six  planks,  each  4m  long,  27.8cm  wide,  and 
75mm  thick,  at  $16  a  hundred  meters. 

48.  Three  sticks  of  timber,  each  8m  long,  22.5cm  wide, 
and  20cm  thick,  at  $17.50  a  hundred  meters. 

49.  A  board  8.25m  long,  28cm  wide  at  one  end  and  35cm 
at  the  other,  and  31.25mm  thick,  at  $0.30  a  meter  ? 

50.  A  stick  of  timber  10m  long,  25cm  thick,  30cm  wide  at 
one  end  and  25cm  wide  at  the  other,  at  $14  a  hundred 
meters. 

51.  The  floor  boards,  32mm  thick,  for  a  two-story  build- 
ing 16m  by  10.5m,  at  $30  a  hundred  meters. 

52.  The  floor  timbers,  25cm  by  50mm,  for  the  building  of 
Ex.  51,  if  the  timbers  run  lengthwise  and  are  placed  on 
edge  30cm  apart,  and  are  worth  $11.50  a  hundred  meters. 

53.  The  fencing  to  enclose  a  field  150m  long  and  75m 
wide  ;  the  posts  are  set  2.5m  apart,  and  cost  $0.25  apiece  ; 
the  fence  is  5  boards  high  ;  the  bottom  board  is  30cm,  the 
top  board  25cm,  and  the  other  three  each  22.5cm  wide,  and 
the  boards  cost  $13.25  a  hundred  meters. 

154.  Kound  logs  are  sold  by  board  measure  after  0.25 
is  deducted  for  slabs. 

Large  and  heavy  timber  is  sold  by  the  ton.  0.20  is  de- 
ducted for  slabs,  and  the  amount  is  reckoned  in  cubic  meas- 
ure instead  of  board  measure. 


76  METRIC    MEASURES. 


Volumes. 

155.  A  solid  has  three  dimensions,  length,  breadth,  and 
thickness  (height  or  depth). 

156.  The  unit  of  volume  is  a  cube,  each  dimension 
of  which  is  a  unit  of  length. 

157.  The  volume  of  a  solid  is  the  number  of  cubic  units 
it  contains. 

158.  The  boundary  of  a  solid  is  called  its  surface.  If 
the  surface  consists  of  planes,  they  are  called  the  faces  of 
the  solid. 

The  faces  of  a  solid  meet  in  lines  called  the  edges  of  the 
solid. 

159.  A  rectangular  solid  is  a  solid  bounded  by  six 
rectangles. 

160.  Find  the  volume   of  a   rectangular   solid  whose 
length  is  5cm,  breadth  3cm,  and  height  7cm. 

The  face  on  which  the  solid  rests  may  be 
divided  into  square  centimeters  ;  there  will 
be  three  rows  of  5qcm  each  ;  in  all  15qcm. 
Upon  each  square  centimeter  may  be 
placed  a  pile  of  7ccm,  so  that  the  solid  will 
contain  15  X  7ccm  ;  that  is,  3  X  5  X  7ccm. 

161.  To  Find  the  Volume  of  a  Rectangular  Solid, 
Express  the  length,  breadth,  and  height  of  the  solid  in  the 

same  linear  unit ;  take  the  product  of  these  numbers  for  its 
volume  in  cubic  units  of  the  same  name  as  the  linear  unit. 

If  the  number  of  cubic  units  in  the  volume  is  divided  by 
the  product  of  the  numbers  of  linear  units  in  any  two  dimen- 
sions, the  quotient  is  the  number  of  linear  units  in  the  third 
dimension. 


METRIC    MEASURES.  77 


Exercise  33. 


1.  How  many  cubic  centimeters  in  a  block  9cm  long, 
7cm  wide,  and  6cm  deep  ? 

2.  If  wood  is  cut  into  120cm  lengths,  and  a  pile  is  43.7m 
long  and  1.4m  high,  how  many  sters  of  wood  are  there  in 
the  pile  ? 

3.  How  many  hektoliters  of  grain  will  a  bin  hold, 
11.2m  long,  4.34m  wide,  and  2.83m  deep  ? 

4.  If  a  liter  of  grain  weighs  0.81  of  the  weight  of  a 
liter  of  water,  find  the  weight  of  the  grain  in  the  bin  of 
Ex.  3. 

5.  A  bin  16m  by  9.7m,  and  2.8m  deep,  is  full  of  oats, 
worth  $0.98  a  hektoliter.     What  is  the  whole  worth  ? 

6.  How  many  liters  does  a  vat  197cm  long,  87cm  wide, 
and  63cm  deep  hold  ?  What  weight  of  water  will  be 
required  to  fill  it  ? 

7.  Add  1341ccm,  2311,  and  2.13M,  and  give  the  sum  in 
terms  of  each  of  the  three  units. 

8.  If  a  spring  delivers  467. 81  each  minute,  how  many 
hektoliters  will  it  deliver  in  60  minutes  ?  in  37  minutes  ? 
in  78  minutes  ? 

9.  If  67. 31  of  oil  in  a  vat  with  perpendicular  sides  fills 
it  to  a  depth  of  173mm,  how  deep  will  13.7  times  that  quan- 
tity fill  it  ?     How  many  hektoliters  will  there  be  ? 

10.  One  cask  contains  171.41  of  oil  ;  another,  209.31 ;  a 
third,  73.81 ;  while  a  square  vat,  137cm  each  way,  is  filled 
to  a  depth  of  69cm.  Eind  in  liters  and  in  hektoliters  the 
amount  of  oil  in  the  four  vessels  together. 

11.  How  many  liters  of  air  in  a  room  7.8m  long,  6.23m 
wide,  and  3m  high  ? 

12.  If  a  person's  breathing  spoils  the  air  at  the  rate  of 
0.2175cbm  a  minute,  how  long  will  it  take  three  persons 
sitting  in  the  closed  room  of  Ex.  11  to  spoil  the  air  ? 


78  METRIC   MEASURES. 

13.  How  long,  at  the  same  rate  as  in  Ex.  11,  will  the  air 
in  a  hall  22m  long,  16m  wide,  and  7m  high  last  280  persons  ? 

162.  To  Find  the  Volume  of  a  Sphere, 

Multiply  the  cube  of  the  diameter  by  0.5236  (£  of  8.1 416). 

14.  How  many  cubic  centimeters  in  a  ball  10cm  in  diam- 
eter? 

15.  Into  a  cubical  box  20cm  on  an  edge,  and  full  of  water, 
an  iron  ball  20cm  in  diameter  is  gently  lowered  until  it 
touches  the  bottom.  Find  in  liters  and  in  cubic  centimeters 
the  volume  of  the  water  left  in  the  box. 

16.  If  cast  iron  weighs  7.207  times  as  much  as  water, 
what  is  the  weight  of  a  cast  iron  ball  5cra  in  diameter? 

17.  A  rubber  ball  is  6.2cm  in  diameter.  What  is  the 
amount  of  rubber  in  the  ball  ? 

18.  If  the  circumference  of  a  cannon  ball  is  52cm,  find 
the  volume  of  the  ball. 

163.  A  cylinder  is  a  solid  bounded  by  two  equal  and 
parallel  circles  called  its  buses,  and  a  uniformly 
curved  surface  called  its  lateral  surface. 

Note.     Two  circles  are  parallel   if   all  points  of 
one  are  equally  distant  from  the  other. 

164.  To  Find  the  Volume  of  a  Cylinder, 
Multiply  the  number  of  square  units  in  its 

base  by  the  number  of  linear  units  hi  its  height. 

19.  How  many  cubic  centimeters  of  oil  are  there  in  a 
cylindrical  cup  10cm  across  when  the  oil  is  38mm  deep? 

20.  What  is  the  capacity  of  a  cylindrical  cup  95mm 
across  and  11.08cm  deep  ? 

21.  What  is  the  capacity  of  a  cylindrical  vessel  16.24cm 
across  and  19.95cm  deep  ?   75.4mm  across  and  87.9nun  deep  ? 


METRIC   MEASURES.  79 

22.  How  many  cubic  meters  of  wood  in  a  round  stick 
of  equal  size  throughout,  37cm  in  diameter  and  8.4m  long  ? 

23.  A  cylindrical  stand-pipe  whose  diameter  is  12m  and 
whose  height  is  22m  is  filled  with  water.  Find  the  weight 
of  the  water. 

24.  Find  the  number  of  liters  of  water  in  a  well,  if  its 
diameter  is  1.2m  and  the  depth  of  the  water  is  2m. 

25.  A  cylindrical  cup  90mm  in  diameter  is  partly  filled 
with  water.  Into  the  cup  is  dropped  a  piece  of  iron,  and 
the  water  rises  63mm.  What  is  the  volume  of  the  piece  of 
iron? 

Exercise  34. 

1.  What  is  the  weight,  in  kilograms,  of  a  hektoliter  of 
water  ?  of  73. 81  of  water  ?  of  a  cubic  meter  of  water  ?  of 
a  cubic  centimeter  of  water  ? 

2.  If  a  man  buys  half  a  ton  of  potatoes  for  $20,  and 
retails  them  all,  without  waste,  at  5  cents  a  kilogram,  what 
profit  does  he  make  on  the  whole  ? 

3.  What  is  the  weight  of  water  required  to  fill  a  vat 
98cm  long,  71cm  wide,  and  38cm  deep  ? 

4.  If  the  vat  of  the  last  example  is  filled  with  brine 
weighing  1.04kg  to  the  liter,  what  is  the  weight  of  the 
brine  ? 

5.  If  the  vat  of  Ex.  3  is  filled  with  wine  weighing 
0.981kg  to  the  liter,  what  is  the  weight  of  the  wine  ? 

6.  What  is  the  total  weight  of  13  men  averaging  73.48kg 
each  ? 

7.  How  many  kilograms,  and  how  many  tons,  will 
3.6175cbm  of  brick  weigh,  at  2  tons  to  a  cubic  meter  ?  at 
2.34  tons  ? 

8.  From  a  barrel  containing  67kg  of  granulated  sugar 
there  are  taken  three  parcels  of  2.75kg  each,  and  four 
parcels  of  7.50kg  each.     How  much  is  left  in  the  barrel  ? 


80  METRIC   MEASURES. 

9.    Into  how  many  pills  of  325mg  each  tan  a  mass  of 
7.8*  be  divided? 

10.  A  mass  of  21.8*  is  divided  into  60  pills.  What  is 
the  weight  of  each  pill  ? 

11.  A  bag,  when  empty,  weighs  213*;  when  full  of 
silver  five-franc  pieces,  20*8  5hg  13*.  A  five-franc  piece 
weighs  25g.    How  many  five-franc  pieces  will  the  bag  hold? 

1 2.  A  vessel,  when  empty,  weighs  2.7kg ;  and  when  full 
of  water  4235dkg.  What  would  it  weigh  if  filled  with 
milk,  which  is  1.03  times  as  heavy  as  water  ? 

Specific  Gravity. 

165.  The  specific  gravity  of  a  substance  is  the  number 
found  by  dividing  the  weight  of  the  substance  by  the 
weight  of  an  equal  bulk  of  water. 

Thus,  if  a  sample  of  quicksilver  has  a  specific  gravity  of  13.6,  it  is 
13.6  times  as  heavy  as  water;  a  cubic  centimeter  of  it  would  weigh 
13.61 ;  a  liter  of  it  would  weigh  13.6k«  ;  and  a  cubic  meter  of  it  would 
weigh  13.6t. 

Again,  if  the  specific  gravity  of  a  certain  alcohol  is  0.827,  that  is, 
if  the  alcohol  weighs  0.827  as  much  as  an  equal  bulk  of  water,  then  a 
cubic  centimeter  of  it  would  weigh  0.827s;  a  liter,  0.827k«;  and  a 
cubic  meter,  0.827*. 

166.  The  specific  gravity  of  a  substance,  therefore,  is 
the  number  that  expresses  the  weight  of  a  cubic  centimeter 
of  it  in  grams  ;  of  a  liter  in  kilograms  ;  of  a  cubic  meter  in 
tons. 

167.  If  a  substance  is  in  water,  the  water  buoys  it  up  by 
just  the  weight  of  the  water  displaced  by  it. 

168.  Examples.  1.  A  lump  of  coal  weighed  in  air  is 
found  to  weigh  1017* ;  weighed  in  water  it  is  found  to 
weigh  321*.     What  is  the  specific  gravity  of  the  coal  ? 


METRIC    MEASURES.  81 

Solution.  1017s  —  321s  =  696s.  Since  the  lump  of  coal  weighs 
696s  less  in  water  than  in  air,  696s  is  the  weight  of  the  water  dis- 
placed by  the  coal,  and  the  lump  contains  696ccm  of  coal. 

Therefore,  the  specific  gravity  of  coal  is  1017s -f- 696s,  or  1.461; 
that  is,  the  number  found  by  dividing  the  weight  of  the  lump  of  coal 
by  the  weight  of  an  equal  bulk  of  water. 

2.  A  stone,  weighing  l.lkg  in  air  and  0.6kg  in  water,  is 
tied  to  a  block  of  wood ;  the  two  together  weigh  1.28kg  in 
air  and  0.54kg  in  water.  What  is  the  specific  gravity  of 
the  wood  ? 

Solution.    The  weight  of  the  wood  in  air  =  1.28ks  —  1.1*8  r=  0.18ks. 

The  weight  of  water  displaced  by  stone  and  wood  =  1.28ks  —  0.54k* 
=  0.74k*. 

The  weight  of  water  displaced  by  stone  alone  =  l.lke—  0.6k«  = 
0.5k«. 

The  weight  of  water  displaced  by  wood,  therefore,  =  0.74ks  —  0.5k« 
=  0.24ks. 

Hence,  the  specific  gravity  of  the  wood  =  0.1 8ks  -f  0.24ks  =  0.76. 


Exercise  35. 

1.  If  a  stone  weighs  1.3kg  in  air  and  0.68kg  in  water,  and 
the  stone  and  a  block  of  wood  together  weigh  l-.55kg  in  air 
and  0.63kg  in  water,  what  is  the  specific  gravity  of  the 
block  of  wood  ? 

2.  What  is  the  weight  of   8.17M  of 
alcohol,  specific  gravity  0.83  ? 

3.  What  will  971  of   alcohol  weigh, 
of   specific    gravity   0.817  ?     of    specific 

Hektoiiter.  gravity     0.819  ?        of     specific    gravity 

0.823  ?     0.838  ?     0.847  ? 

4.  A  bar  of  aluminum  113mm  long,  17mm  wide,  and 
13mra  thick  is  said  to  be  of  specific  gravity  2.57.  What 
does  it  weigh  ?  If  it  really  is  of  specific  gravity  2,67, 
what  does  it  weigh  ? 


82  METRIC    MEASURES. 

5.  What  would  be  the  specific  gravity  of  the  aluminum 
in  Ex.  4  if  the  bar  weighed  65.1378  ? 

6.  What  is  the  weight  of  a  bar  of  aluminum  371mm  by 
63mm  by  84mm,  specific  gravity  being  2.63  ? 

7.  An  irregular  mass  of  copper,  gently  lowered  into  a 
pail  brimful  of  water,  caused  1.3741  to  run  over.  What 
did  it  weigh  if  of  specific  gravity  8.91  ?  if  8.89  ? 

8.  What  would  be  the  specific  gravity  of  the  copper  in 
Ex.  7  if  the  mass  weighed  12.3016k«  ? 

169.   To  Find  the  Specific  Gravity  of  a  Substance, 

Divide  the  weight  in  grants  by  the  bulk  in  cubic  centi- 
meters, the  weight  in  kilograms  by  the  bulk  in  liters,  or  the 
weight  in  tons  by  the  bulk  in  cubic  meters. 

9.  A  plate  of  iron  137cm  long,  64.3cm  wide,  and  4.31cm 
thick  weighs  277.54k«.  What  is  its  specific  gravity? 
What  would  the  same  mass  weigh  at  specific  gravity  7.47  ? 
at  7.79  ? 

10.  What  is  the  specific  gravity  of  sea  water  when  a 
hektoliter  weighs  102.58k«  ?     when  31  weighs  3077*  ? 

11.  What  is  the  specific  gravity  of  a  substance  of  which 
7.3ccm  weighs  31.5*  ? 

12.  If  a  cubic  meter  of  sand  weighs  1723kg,  what  is  its 
specific  gravity  ?  If  3.4cbm  of  gravel  weighs  7.134  tons, 
what  is  its  specific  gravity  ? 

13.  If  a  cubic  centimeter  of  metal  weighs  7.3g,  what  is 
its  specific  gravity  ? 

14.  What  is  the  specific  gravity  of  a  fluid  weighing 
2.317k«  to  a  liter  ? 

15.  If  a  body  weighs  3.71k*  in  air  and  2.38k*  in  water, 
what  is  its  specific  gravity  ? 

16.  A  piece  of  ore  weighing  3.77kg  weighs  in  water  only 
2.53kg.     What  is  its  specific  gravity  ? 


METRIC    MEASURES.  83 

17.  How  many  cubic  centimeters  in  a  stone  which  loses 
17.8g  of  its  weight  when  weighed  in  water  ?  What  is  its 
specific  gravity  if  it  weighs  33.7g  in  air  ? 

18.  In  a  wrought-iron  bottle  I  find  2.631  of  quicksilver, 
weighing  35.81kg  ;  in  another,  2.591,  weighing  35.193kg  ;  in 
a  third,  2.6171,  weighing  35.571kg  What  is  the  specific 
gravity  of  each  ?  What  would  be  the  specific  gravity  of 
the  mixture  if  the  three  were  emptied  into  one  vessel  ? 

19.  A  plate  of  iron  89cm  by  17cm  by  7cm  weighs  79.43kg. 
What  is  its  specific  gravity  ? 

20.  What  is  the  specific  gravity  of  a  rectangular  block  of 
wood  1.6m  long,  0.3m  wide,  and  0.15m  thick,  if,  floating  in 
water  on  its  face  0.3m  wide,  it  sinks  to  a  depth  of  0.12m  ? 

Exercise  36. 

1.  If  3  men  eat  8kg  of  bread  a  week,  how  much  will  1 
man  eat  at  the  same  rate  ?  How  much  will  7  men  ?  How 
much  will  the  3  men  eat  in  1  day  ?  How  much  will  1  man 
eat  in  1  day  ?  How  much  will  7  men  eat  in  1  day  ?  in  1 
week  ?  in  5  weeks  ? 

2.  At  the  same  rate  as  in  Ex.  1,  how  much  will  17 
men  eat  in  3  weeks  and  4  days  ? 

3.  If  1M  of  oats  is  enough  for  5  horses  1  week,  how 
much  is  enough  for  1  horse  1  week  ?  for  1  horse  7  weeks  ? 
for  11  horses  17  weeks  ? 

4.  If  2hl  of  grain  is  enough  for  3  horses  5  days,  how 
much  is  enough  for  3  horses  1  day  ?  for  1  horse  1  day  ? 
for  7  horses  6  days  ? 

5.  Mix  171  of  vinegar,  costing  6  cents  a  liter,  with 
391  at  5  cents,  211  at  7  cents,  and  131  of  water  costing  noth- 
ing.    Find  the  number  of  liters  and  the  cost. 

6.  For  how  much  a  liter  must  I  sell  the  mixture  of 
of  Ex.  5  to  gain  96  cents  ?  to  gain  $1.41  ? 


84  METRIC    MEASURES. 

7.  A  grocer  sold  421  kegs  of  butter  for  $4995.25 ;  56 
kegs  brought  $12.50  a  keg,  91  brought  $11.75  a  keg,  and 
100  kegs  brought  $12.25  a  keg.  For  how  much  a  keg 
were  the  other  kegs  sold  ? 

8.  If  3  tons  of  coal  cost  $15.75,  how  many  tons  will 
$36.75  buy  ? 

9.  If  5ra  of  cloth  cost  $18.75,  what  will  7m  cost  ? 

10.  If  a  tap  running  3.51  a  minute  fills  a  tub  in  16 
minutes,  how  long  will  a  tap  delivering  51  a  minute  be 
in  filling  the  same  tub  ? 

11.  If  both  taps  of  the  last  example  are  opened  at  once, 
how  soon  will  they  fill  the  tub  ? 

12.  If  3  men  can  dig  378m  of  ditch  in  2  days,  how  long 
will  it  take  5  men,  at  the  same  rate,  to  dig  787m  ? 

13.  Into  a  tub  that  will  hold  481  one  tap  is  delivering 
water  at  the  rate  of  3.71  a  minute  ;  while  out  of  it,  by  an- 
other tap,  the  water  is  running  at  2.51  a  minute.  How 
long  will  it  take  to  fill  the  tub,  beginning  with  it 
empty  ? 

14.  A  tap  discharges  into  a  tub  4.21  a  minute  ;  from  the 
tub  water  is  also  running,  by  a  second  tap  ;  the  water  in 
the  tub  gains  301  in  18  minutes.  How  fast  is  the  second 
tap  discharging  ? 

15.  If  a  wheel  is  1.2m  across,  how  many  times  will  it 
turn  in  going  one  kilometer  ? 

16.  How  many  times  in  a  minute  does  the  wheel  of  the 
last  example  turn,  when  the  carriage  is  driven  at  the  rate 
of  14km  an  hour  ? 

17.  What  is  the  weight  of  the  water  in  a  tank  if  it 
takes  1  hour  and  38  minutes  at  the  rate  of  8.71  a  minute 
to  empty  the  tank  ? 

18.  If  we  replace  the  water  of  Ex.  17  with  oil  worth 
$18.75  a  hektoliter,  what  will  the  contents  of  the  tank  be 
worth  ? 


METRIC    MEASURES.  85 


Exercise  37. 

1.  A  train  leaves  Paris  at  11  o'clock  a.m.  and  reaches 
Lyons  at  10  o'clock  p.m.  How  many  meters  does  it  travel 
in  an  hour,  the  distance  from  Paris  to  Lyons  being  512. 7km  ? 

2.  A  railroad  has  a  single  track  11.450km  long.  How 
many  rails  4.569m  in  length  did  it  require  to  lay  the 
track  ? 

3.  A  book  is  2.1cm  in  thickness  ;  each  leaf  is  0.05mm 
thick.     Find  the  number  of  pages  in  the  book. 

4.  If  the  cost  of  opening  a  canal  amounts  to  $25,400 
a  kilometer,  how  much  will  a  canal  cost  which  is  113.253km 
in  length  ? 

5.  The  expense  of  laying  out  a  paved  road  is  $12,500 
a  kilometer.  How  much  will  a  road  cost  which  is  72.053km 
long? 

6.  The  cost  of  building  a  railroad  is  about  $78,000  a 
kilometer  in  France,  and  only  $25,000  in  the  United 
States.  How  much -will  it  cost  in  each  country  to  make 
a  road  295.671km  long  ? 

7.  If  you  must  go  up  211  steps  to  reach  the  top. of  a 
tower,  and  each  step  is  195mm  high,  what  is  the  height  of 
the  tower  ? 

8.  A  house  has  5  stories,  each  story  has  19  stairs,  each 
stair  is  16cm  in  height.  Find  the  height  of  the  floor  of 
the  fifth  story  from  the  ground. 

9.  A  ream  of  paper  contains  20  quires,  each  quire  has 
24  sheets,  the  ream  is  13.5cm  in  thickness.  Find  the  thick- 
ness of  each  sheet. 

10.  The  equator  on  a  terrestrial  globe  measures  0.80m 
in  circumference.  By  the  aid  of  a  tape  measure  we  find 
that  the  distance  between  two  cities  on  this  globe  is  0.046'11. 
What  is  really  the  distance  in  kilometers  between  the  two 
cities  ?     (The  earth's  equator  is  40,075.45km.) 


86  METRIC    MEASURES. 

1 1 .  Upon  a  military  map  we  find  that  the  distance  from 
Paris  to  St.  Denis  is  78mm.  What  is  the  distance  in  kilo- 
meters from  Paris  to  St.  Denis  ?  The  map  is  made  on  the 
scale  of  1  to  80,000  ;  that  is,  lm  on  the  map  represents 
80,000m  of  actual  measurement  upon  the  ground. 

12.  Find  the  number  of  revolutions  made  by  the  wheels 
of  a  carriage  in  traveling  82*".  The  wheels  are  lSo^""11 
in  diameter. 

13.  How  many  hektars  in  a  square  kilometer?  how 
many  ars  ?  how  many  square  meters  ? 

14.  France  has  about  542,000qkm.  How  many  hektars 
does  it  measure  ? 

15.  A  piece  of  land  1224.5m  square  is  sold  at  $140  a 
hektar.     How  much  does  the  land  bring  ? 

16.  The  total  surface  measurement  of  the  glass  in  the 
windows  of  a  house  is  182qm.  How  many  panes  of  53cm 
by  48cra  will  it  take  to  supply  the  windows  ? 

17.  How  many  square  slabs  of  marble  150qcm  on  the 
surface  will  it  require  to  pave  a  court  whose  area  is 
25.35qm  ? 

18.  A  speculator  bought  31.0728ha  of  land  for  $1296  a 
hektar.  For  how  much  a  square  meter  must  he  sell  it  to 
realize  a  profit  of  $1937  ? 

19.  A  man  is  offered  $6000  for  2.5a  of  land.  He  de- 
clines to  sell ;  and  soon  after  the  town  gives  him  $25.20  a 
square  meter.  How  much  did  he  make  by  refusing  the 
first  offer  ? 

20.  A  man  surveys  a  piece  of  land  and  finds  that  it 
measures  14.0715ha.  He  afterwards  discovers  that  his 
chain  was  too  short  by  0.03m.  How  can  he  calculate  the 
real  superficial  measurement  of  the  land  without  surveying 
it  again  ?     (A  surveyor's  chain  is  10m  long.) 

21.  A  pile  of  wood  is  4.25m  long,  1.33m  thick,  and  2.60m 
high.     How  many  sters  are  there  in  it  ? 


METRIC    MEASURES.  87 

22.  The  railroad  from  Paris  to  Orleans  has  a  double 
track ;  each  rail  is  4m  long,  and  the  distance  from  Paris  to 
Orleans  is  121km.  What  is  the  number  of  rails  used  in 
laying  the  track  ?  If  the  width  of  the  road  is  15m,  how 
many  hektars  of  land  does  the  road  include  ? 

23.  Find  the  number  of  ars  in  a  surface  which  a 
ream  of  paper  (480  sheets)  will  cover.  The  sheets  are 
30.3cm  long  and  195mm  wide. 

24.  A  beam  is  7.070m  long  ;  its  two  other  dimensions 
are  0.258m  and  87mm.     Find  its  volume. 

25.  A  bar  of  iron  3m  long  measures  45mm  square  on  the 
end  where  it  has  been  evenly  cut.  The  bar  is  heated  and 
drawn  out  to  a  greater  length  by  being  passed  through  an 
orifice  24mm  square.  What  is  the  length  of  the  bar  after 
the  operation  ? 

26.  A  reservoir  is  1.50m  wide,  2.80m  long,  and  1.25m 
deep.  Find  how  many  liters  it  contains  when  full,  and  to 
what  height  it  would  be  necessary  to  raise  it  that  it  might 
contain  10cbm. 

27.  Suppose  a  box  to  be  3.75m  long,  3.50m  wide,  and 
0.50m  high.  How  much  lime  would  it  take  to  fill  it  with 
mortar,  reckoning  that  lcbm  of  lime  after  being  slaked  be- 
comes 1.80cbm  of  mortar  ? 

28.  A  chest  has  the  following  dimensions  :  1.17m,  0.90m, 
1.04m.  If  0.12  of  the  volume  of  the  chest  is  deducted  for 
packing,  how  many  cakes  of  soap  13cm  square  on  the 
bottom  and  29cm  thick  could  be  put  in  it  ? 

29.  A  cubic  meter  of  dry  plaster  makes  1.18cbm  when 
tempered ;  tempered  plaster  increases  1  in  every  100 
twenty-four  hours  after  it  is  mixed.  What  volume  of 
tempered  plaster  would  be  obtained  from  55  sacks  of  251 
each  of  dry  plaster  ? 

30.  A  reservoir  is  2.80m  long,  1.50m  wide,  and  1.25m 
deep.     How  many  liters  will  be  required  to  fill  0.80  of  it  ? 


88  METRIC    MEASURES. 

31.  A  man  buys  1415m  of  wheat  for  $3.50  a  hektoliter ; 
but  the  measure  used  proves  too  small,  the  mistake 
amounting  to  31  in  every  hektoliter.  What  was  the  quan- 
tity of  wheat  delivered  to  the  purchaser,  the  cost,  and  the 
reduction  which  ought  to  be  made  to  him  on  account  of 
the  error  ? 

32.  The  dimensions  of  a  tile  are  as  follows  :  length 
22cm,  width  llcm,  thickness  55mm.  Find  the  volume  of 
the  tile,  and  the  number  of  tiles  in  a  pile  of  25cbm. 

33.  The  measurement  of  a  pile  of  wood  shows  that  a 
ster  could  be  filled  from  it  25.68  times.  Find  the  volume 
of  the  pile  in  cubic  meters,  reckoning  the  length  of  the 
logs  to  be  1.1 5m. 

Note.  A  ster  is  a  frame,  as  represented  on  page  64,  one  meter 
high  and  one  meter  between  the  upright  posts.  The  ster  may  be 
filled  with  wood  of  any  length,  and  the  volume  will  be  as  many 
cubic  meters  as  the  sticks  of  wood  are  meters  long. 

34.  A  liter  of  air  weighs  1.273*.  How  much  does  a 
cubic  meter  of  air  weigh  ?  How  many  times  as  heavy  as 
air  is  water  ? 

35.  A  package  of  candles  that  weighs  465g  is  sold  for 
28  cents.  At  the  same  rate  what  is  the  price  of  a  kilogram 
of  candles  ? 

36.  How  many  times  will  3.243*  of  water  fill  a  liter 
measure  ? 

37.  Express  in  kilograms  the  weight  of  43.4578ccm  of 
pure  water. 

38.  The  volume  of  the  axle  of  an  engine  is  0.245cbm. 
Find  its  weight,  if  the  specific  gravity  of  the  iron  is  7.8. 

39.  Find  the  volume  of  a  gram  of  the  following  sub- 
stances :  proof  spirit,  specific  gravity  0.865 ;  tin,  specific 
gravity  7.291  ;  lead,  specific  gravity  11.35 ;  copper,  spe- 
cific gravity  8.85  ;  silver,  specific  gravity  10.47 ;  cork, 
specific  gravity  0.240. 


METRIC   MEASURES.  89 

40.  Olive  oil  costs  60  cents  a  kilogram.  What  is  the 
price  of  a  liter  ?     The  specific  gravity  of  olive  oil  is  0.914. 

41.  Pure  alcohol  costs  $1.87  a  kilogram.  What  is  the 
price  of  a  liter  ?     The  specific  gravity  of  alcohol  is  0.792. 

42.  A  man  wishes  to  build  a  shed  large  enough  to  hold 
1358t  of  wood  ;  if  the  shed  is  to  be  3m  high  and  5m  wide, 
how  long  must  it  be  ? 

43.  In  a  country  where  firewood  is  cut  1.16m  long,  what 
must  be  the  height  of  the  sides  of  the  ster  that  it  may  hold 
a  cubic  meter  ? 

44.  If  a  ster  of  cork  costs  $20.00,  how  much  would 
100kg  cost,  the  cork  weighing  0.25  as  much  as  water  ? 

45.  A  liter  of  powder  weighs  825g.  What  will  be  the 
volume  in  cubic  centimeters  of  a  charge  for  a  gun  if  the 
charge  weighs  5g  ? 

46.  Out  of  gold  which  weighs  19.362  times  as  much  as 
water  sheets  of  gold  foil  are  made  which  are  0.010mm  in 
thickness.     What  surface  will  3g  of  gold  cover  ? 

47.  Find  the  weight  of  an  oak  board  3.25m  long,  0.31ra 
wide,  and  0.04m  thick,  if  the  specific  gravity  of  the  oak 
is  0.808. 

48.  Find  the  weight  of  a  bar  of  iron  having  the  follow- 
ing dimensions  :  length  3.6m,  width  6cm,  thickness  2cm,  if 
the  specific  gravity  of  the  iron  is  7.8. 

49.  How  many  lead  balls,  each  weighing  27g,  can  be 
obtained  by  melting  a  cubic  mass  of  lead  0.356m  on  an 
edge,  if  the  specific  gravity  of  the  lead  is  11.35  ? 

50.  Marble  costs  $ 30.95  a  cubic  meter,  and  the  specific 
gravity  of  marble  is  2.73.  If  a  block  of  marble  weighs 
1260kg,  what  is  its  volume  and  what  is  it  worth  ? 

51.  Sea  water  contains  28  parts,  by  weight,  of  salt  in 
1000.  A  liter  of  sea  water  weighs  1.025kg.  How  many 
kilograms  of  salt  can  be  obtained  from  126.276842cbm  of 
sea  water  ? 


90  METRIC    MEASURES. 

52.  An  empty  cask  weighs  17.06kg  ;  when  filled  with 
water  it  weighs  275. 8kg.  How  many  liters  does  it  hold  ? 
How  many  casks  of  this  size  will  it  take  for  the  wine 
from  a  vat  containing  3.008cbm? 

53.  If  it  takes  2.048m  of  wheat  to 
sow  a  hektar,  how  many  cubic  meters 
will  it  take  to  sow  a  square  kilo- 
meter ? 

54.  A  piece  of  road  lkm  long  and 
7m  wide  is  to  be  macadamized  to  the 
depth  of  33cm.     What  will  the  work 

Hektoiiter.  cost  at  43  cents  a  cubic  meter  ? 

55.  A  gasometer  holds  28,000cbm  of  gas.  How  many 
jets  will  this  gasometer  feed  for  an  evening,  when  each  jet 
burns  1251  an  hour,  and  is  used  4  hours  ? 

56.  The  city  of  Venice  is  situated  in  the  midst  of  a 
great  lake  of  salt  water,  communicating  with  the  sea,  and 
all  the  rain  water  is  caught  for  the  cisterns.  Ordinary 
years  the  fall  of  rain  in  Venice  is  82cm  ;  the  surface  of  the 
city,  after  the  canals  have  been  deducted,  is  520** .  Reck- 
oning the  population  at  115,530,  how  many  liters  a  day  of 
rain  water  can  each  inhabitant-  have  ? 

57.  Find  the  weight  of  a  bar  of  iron  5.35m  long,  4.56cm 
thick,  and  3.54cm  wide.  Find,  also,  the  width  of  an  oak 
beam  4.30m  long,  9.12cm  thick,  which  has  the  same  weight. 
The  specific  gravity  of  the  oak  to  be  reckoned  at  1.026, 
that  of  the  iron  7.788. 

58.  Find  the  specific  gravity  and  volume  of  a  body 
weighing  35kg  in  air  and  30kg  in  water. 

59.  A  ster  of  piled  oak  wood  weighs  425kg;  the  specific 
gravity  of  the  wood  is  0.74.  What  is  the  volume  occupied 
by  the  spaces  between  the  logs  ?  For  how  much  must 
lOO1*  of  separate  sticks  be  sold  to  bring  the  same  amount 
as  when  sold  at  $2.20  a  ster  ? 


METRIC   MEASURES.  91 

60.  Wrought  iron  sells  for  $7.00  per  100kg.  A  bar  of 
iron  4.5cm  wide,  3.3cm  thick  costs  $5.08  ;  what  is  its 
length,  reckoning  the  specific  gravity  of  the  iron  at  7.4  ? 

61.  Experiment  shows  that  water  weighs  770  times  as 
much  as  air  ;  and  the  specific  gravity  of  mercury  is  13.6. 
How  many  liters  of  air  will  it  take  to  weigh  as  much  as  a 
liter  of  mercury  ? 

62.  A  mass  of  lead  weighing  753kg  is  made  into  sheets 
0.1mm  thick.  Find,  in  square  meters,  the  surface  which 
can  be  covered  by  the  sheets  thus  obtained.  The  specific 
gravity  of  the  lead  is  11.3. 

63.  A  rectangular  sheet  of  tin  of  uniform  thickness  is 
85cm  wide,  1.35m  long,  and  weighs  268g.  What  is  its  thick- 
ness, if  the  specific  gravity  of  tin  is  7.3  ? 

64.  The  fine  coal  which  collects  about  the  shafts  of  the 
mines  and  in  the  coalyards  was  for  a  long  time  wasted, 
because  it  could  not  be  burned  in  stoves  and  grates.  Now 
this  dust  is  mixed  with  tar  in  the  proportion  of  92kg  of 
dust  and  8kg  of  tar  ;  Jihe  mixture  is  heated,  and  afterwards 
pressed  in  rectangular  moulds  14.75cm  by  18.5cm  by  29cra  ; 
each  one  of  these  blocks  weighs  10kg.  They  are  sold  at 
$3.00  a  ton,  and  make  excellent  fuel  for  heating  steam 
boilers.  Find  the  specific  gravity  of  this  fuel ;  also  the 
sum  which  would  be  realized  in  thus  utilizing  800,000*  of 
coal  dust,  the  cost  of  tar,  mixing,  etc.,  being  $0.50  a  ton. 

65.  A  bar  of  iron  a  millimeter  square  on  the  end  will 
break  under  a  tension  of  30kg.  Find  the  length  at  which  a 
suspended  bar  of  iron  will  break  from  its  own  weight,  if 
the  specific  gravity  of  the  iron  is  7.8. 

66.  Fifty-three  kilograms  of  starch  are  obtained  from 
100kg  of  wheat.  A  hektar  of  land  produces  13631  of  wheat ; 
a  hektoliter  of  wheat  weighs  78kg.  If  the  wheat  harvested 
from  a  field  measuring  2ha  and  33qm  is  taken  to  a  starch 
factory,  how  much  starch  will  be  made  from  it  ? 


92  METRIC    MEASURES. 

67.  A  gardener  wishes  to  provide  glass  for  his  hotbeds. 
The  beds  cover  2.65a;  the  panes  will  cover  0.75  of  the 
whole  surface,  the  rest  being  taken  up  by  the  frames  and 
alleys.  First  find  how  many  panes  measuring  45cm  by  37cm 
it  will  take  to  cover  the  beds  ;  then  find  the  price  of  the 
glass,  at  a  cost  of  95  cents  a  square  meter. 

68.  A  jar  full  of  water  weighs  1.325**;  filled  with  mer- 
cury it  weighs  12.540kg.  Find  the  capacity  and  the  weight 
of  the  jar,  if  the  specific  gravity  of  the  mercury  is  13.59. 

69.  A  hektoliter  of  rape  seed  weighs  63**,  and  321  of  oil 
can  be  extracted  from  it.  How  many  kilograms  of  the 
seed  will  it  take  to  make  a  hektoliter  of  oil  ? 

70.  Common  burning  gas  is  0.97  of  the  weight  of  air, 
and  a  liter  of  air  weighs  1.293g.  In  a  shop  there  are  65 
jets,  each  one  of  which  burns  1231  an  hour,  and  is  used  5 
hours  in  the  winter  evenings.  Find  the  weight  of  the  gas 
used  in  a  month  of  26  days,  and  the  expense  of  lighting 
the  shop,  when  gas  costs  6  cents  a  cubic  meter. 

71.  A  merchant  buys  one  kind  of  wine  at  30  cents  a 
liter,  another  kind  at  21  cents  a  liter ;  he  mixes  the  two 
kinds  by  putting  51  of  the  first  with  8l  of  the  second.  For 
how  much  a  liter  must  he  sell  the  mixture  in  order  to  gain 
$3.75  a  hektoliter  ? 

72.  If  it  requires  360  tiles  to  drain  an  ar  of  land,  what 
will  it  cost  to  drain  17.784ha,  when  the  tiles  cost  $20  a 
thousand,  and  the  expense  of  laying  is  the  same  as  the 
cost  of  the  tiles  ? 

73.  Hewn  stone  of  medium  durability  ought  not  to 
support,  as  a  permanent  weight,  more  than  0.07  of  the 
weight  that  is  required  to  crush  it.  A  certain  kind  of 
stone  used  for  building  will  be  crushed  under  a  weight  of 
250**  a  square  centimeter.  What  is  the  greatest  height  to 
which  a  wall  constructed  of  this  material  can  be  safely 
carried,  if  the  specific  gravity  of  the  stone  is  2.1  ? 


METRIC    MEASURES.  93 

74.  Several  different  kinds  of  wine  are  mixed  as  follows  : 
2451  at  20  cents  a  liter,  5471  at  15  cents  a  liter,  3441  at  25 
cents  a  liter.     How  much  does  the  mixture  cost  a  liter  ? 

75.  A  farmer  wishes  to  drain  a  field  of  8.75ha.  Each 
hektar  requires  750m  of  ditches.  The  opening  of  these 
ditches  costs  10  cents  a  running  meter  ;  the  tiles  are  30cm 
long  and  cost  $15  a  thousand.  He  pays  2  cents  a  meter 
for  laying  the  tiles,  and  4  cents  a  meter  for  filling  the 
ditches.     What  is  the  cost  of  draining  the  field  ? 

76.  A  silver  five-franc  piece  weighs  25g,  and  is  com- 
posed of  9  parts  of  pure  silver  and  1  part  of  pure  copper. 
A  silver  two-franc  piece  weighs  10g,  and  is  composed  of  835 
parts  of  pure  silver  and  165  parts  of  pure  copper.  A 
silver  twenty -centime  piece  weighs  lg,  and  has  the  same 
composition  as  the  two-franc  piece.  Find  the  total  weight 
of  pure  silver  and  of  pure  copper  contained  in  272  five- 
franc  pieces,  145  two-franc  pieces,  and  179  twenty-centime 
pieces. 

77.  The  dimensions  of  the  bottom  of  a  rectangular  box 
are  70cm  by  50cm.  If  the  box  contains  exactly  an  hektoliter 
of  wheat  when  full,  what  is  the  height  of  the  box  ? 

78.  If  a  stick  of  oak  timber.  54  centimeters  wide  and  65 
centimeters  thick  costs  $25  at  $16  a  cubic  meter,  what  is 
the  length  of  the  stick  ? 

79.  A  rectangular  box  whose  bottom  is  a  square  28cm  on 
a  side,  and  whose  height  is  19.2cm,  is  exactly  filled  with 
gold  twenty-franc  pieces,  in  piles  touching  each  other.  If 
a  twenty-franc  piece  is  35mm  in  diameter,  and  1.28mm  thick, 
what  is  the  value  of  the  gold  in  the  box  ? 

80.  If  lm  of  coal  yields  1854cbm  of  gas,  and  one  burner 
consumes  1401  of  gas  in  an  hour,  how  many  hektoliters  of 
coal  are  required  to  supply  2800  burners  for  144  hours  ? 

8 1 .  How  many  liters  of  water  in  a  cylindrical  well  1.96m 
in  diameter,  if  the  water  is  2.84ra  deep  ? 


CHAPTER  VI. 

MEASURES  AND  MULTIPLES  OF  NUMBERS. 

170.  Factors  of  a  Number.  The  factors  of  a  number 
are  the  numbers  whose  product  is  that  number. 

171.  Prime  Numbers.  A  prime  number  is  a  number 
that  has  no  integral  factors ,  except  itself  and  one. 

Thus,  2,  3,  5,  7,  11,  13,  17,  19  are  prime  numbers. 

172.  Composite  Numbers.  A  composite  number  is  a 
number  that  is  the  product  of  two  or  more  integral  factors. 

Thus,  10,  21, 143  are  composite  numbers,  for  10  is  2  X  6  ;  21  is  3  X 
7;  143  is  11  X  13. 

Note.  In  speaking  of  the  integral  factors  of  a  number  we  exclude 
the  number  itself  and  one. 

173.  Prime  Factors.  A  prime  factor  is  a  factor  that  is 
a  prime  number. 

174.  A  composite  number  can  have  but  one  set  of  prime 
factors. 

Thus,  12  cannot  be  expressed  as  the  product  of  any  set  of  prime 
factors  except  2X2X3.  It  is  the  product  of  2  X  6,  and  of  3  X  4, 
but  one  of  the  factors  of  2  X  6  and  one  of  3  X  4  is  composite. 

175.  A  number  that  can  be  divided  by  another  without 
a  remainder  is  said  to  be  exactly  divisible  by  that  number  ; 
and  the  divisor  is  called  an  exact  divisor. 

176.  Even  Numbers.  An  even  number  is  a  number  that 
is  exactly  divisible  by  2. 

177.  Odd  Numbers.  An  odd  number  is  a  number  that 
is  not  exactly  divisible  by  2. 


MEASURES    AND    MULTIPLES    OF    NUMBERS.  95 

178.  To  Determine  Prime  Numbers.  Write  the  series 
of  integral  numbers  in  order,  beginning  with  the  smallest ; 
then  cancel  the  even  numbers,  and  place  a  dot  over  each 
multiple  of  3  :  we  have 

1,  %  3,  4,  5,  $,  7,  $,  9,  Xd>,  11,  it,  13,  U>  15,  X$,  17,  X$} 

19,  n,  21,  U,  23,  U,  25,  M,  27,  U,  29,  $>,  31,  etc. 

Each  multiple  of  6  is  cancelled,  and  also  has  the  dot 
over  it,  and  the  only  numbers  without  the  cancelling  line 
or  the  dot  come  just  before  or  just  after  a  multiple  of  6. 
Therefore, 

The  only  numbers  greater  than  6  that  can  be  prime  num- 
bers are  one  less  than  or  one  greater  than  a  multiple  of  6. 

179.  Examples.     1.    Find  the  prime  factors  of  144. 


144 

_72 
_36 

9 
3 


That  is,  144  =  2  X  2  X  2  X  2  X  3  X  3  =  24  X  32. 

Note.  Divide  as  many  times  as  possible  by  the  smallest  prime 
number  that  will  exactly  divide  the  given  number  before  taking  the 
next  larger  prime  number  for  a  divisor. 

2.    Find  the  prime  factors  of  233. 

By  actual  trial  we  find  that  the  prime  numbers  2,  3,  5,  7,  11,  13, 
and  17  are  not  factors  of  233.  We  need  not  try  any  higher  prime 
number  than  17,  as  the  quotient  when  17  is  tried  is  less  than  17. 
Therefore,  no  prime  number  greater  than  17  can  be  a  factor  ;  and  we 
have  found  by  trial  that  no  prime  number  less  than  17  is  a  factor. 
Therefore,  233  is  a  prime  number. 

Notice  that  233  is  one  less  than  234,  or  39  X  6. 


96         MEASURES    AND    MULTIPLES    OF    NUMBERS. 

180.  From  these  two  examples  we  have  the  following 

Rule.  Divide  the  given  number  by  any  prime  number 
that  exactly  divides  it ;  then  the  quotient  by  any  prime  nit  tu- 
ber that  exactly  divides  it ;  and  so  on  until  the  quotient  is 
itself  a  prime  number.  The  several  divisors  and  the  last 
quotient  are  the  prime  factors. 

If  no  prime  factor  is  found  before  the  quotient  becomes 
equal  to  or  less  than  the  divisor,  the  number  is  a  pHme 
number. 

181.  The  following  tests  are  very  useful  for  determining 
without  actual  division  whether  a  number  contains  certain 
factors  : 

1.  A  number  is  divisible  by  2  if  its  last  digit  is  even. 

2.  A   number   is    divisible   by  4    (22)   if  the  number 
denoted  by  the  last  two  digits  is  divisible  by  4. 

3.  A   number   is   divisible   by  8  (2s)  if   the  number 
denoted  by  the  last  three  digits  is  divisible  by  8. 

4.  A  number  is  divisible  by  3  if  the  sum  of  its  digits  is 
divisible  by  3. 

5.  A  number  is  divisible  by  6  if  its  last  digit  is  even 
and  the  sum  of  its  digits  is  divisible  by  3. 

6.  A  number  is  divisible  by  9  (32)  if  the  sum  of  its 
digits  is  divisible  by  9. 

7.  A  number  is  divisible  by  5  if  its  last  digit  is  either 
5  or  0. 

8.  A  number  is   divisible   by  25  (5*)  if   the  number 
denoted  by  the  last  two  digits  is  divisible  by  25. 

9.  A  number  is  divisible  by  125  (58)  if  the  number 
denoted  by  the  last  three  digits  is  divisible  by  125. 

10.  A  number  is  divisible  by  11  if  the  difference  between 
the  sum  of  the  digits  in  the  even  places  and  the  sum  of  the 
digits  in  the  odd  places  is  either  0  or  a  multiple  of  11. 


MEASURES    AND    MULTIPLES    OF    NUMBERS. 


97 


182.  Other  prime  factors,  7,  13,  17,  19,  etc.,  sometimes 
betray  their  presence  to  one  familiar  with  the  subject; 
but,  practically,  the  best  way  to  detect  them  is  by  actual 
division. 

183.  If  we  divide  any  number  less  than  121  (ll2)  by 
11,  or  by  a  number  greater  than  11,  it  is  plain  that  the 
quotient  is  less  than  11. 

If  we  divide  any  number  between  121  and  143  (11  X  13) 
by  11,  the  quotient  will  evidently  lie  between  11  and  13  ; 
and,  since  there  are  no  prime  numbers  between  11  and 
13,  the  quotient,  if  a  whole  number,  must  be  composite, 
and  contain  factors  smaller  than  11. 

What  is  true  of  11  and  13  is  evidently  true  of  any  two 
adjacent  prime  numbers  ;  namely,  that,  excepting  the 
second  power  of  the  smaller  prime,  every  composite  number 
less  than  the  product  of  two  adjacent  prime  numbers  contains 
prime  factors  less  than  the  smaller  of  these  two  numbers. 

Thus,  every  composite  number  less  than  4087  (61  X  67),  except 
3721  (612),  contains  prime  factors  less  than  61. 

184.  The  value  of  the  following  table,  in  discovering 
the  prime  factors  of  a  given  number,  will  be  apparent. 


Primes  .  . 

7 

11 

13 

17 

19 

23 

29 

31 

37 

Powers  .  . 

49 

121 

169 

289 

361 

529 

841 

961 

1369 

Products  . 

77 

143 

221 

323 

437 

667 

899 

1147 

1517 

Primes  .  . 

41 

43 

47 

53 

59 

61 

67 

71 

73 

Powers  .  . 

1681 

1849 

2209 

2809 

3481 

3721 

4489 

5041 

5329 

Products  . 

1763 

2021 

2491 

3127 

3599 

4087 

4757 

5183 

5767 

Primes  .  . 

79   83 

89 

97 

101 

103 

107 

109 

113 

Powers  .  . 

6241 

6889 

7921 

9409 

10201 

10609 

11449 

11881 

12769 

Products  . 

6557 

7387 

8633 

9797 

10403  11021 

11663 

12317 

14351 

Opposite  ' '  Powers ' '  are  placed  the  squares  of  the  primes  from  7  to 
113;  and  opposite  "Products"  are  placed  the  products  of  the  suc- 
cessive pairs  of  adjacent  primes  from  7  to  127. 


22,610,764 

3 

152,691 

7 

50,897 

11 

7,271 

98         MEASURES    AND    MULTIPLES    OF    NUMBERS. 

185.    Example.     Find  the  prime  factors  of  610,764. 

As  64  is  divisible  by  4,  but  764  is  not  divisible  by  8,  22  is  the  high- 
est power  of  2  contained  in  610,764. 

As  the  sum  of  the  digits  152,691  is  divisible  by  3, 
but  not  by  9,  31  is  the  highest  power  of  3  contained  in 
152,691. 

The  next  quotient,  50,897,  does  not  contain  5  ;  but 
661     divided  by  7  gives  7271.     7271  does  not  contain  7; 
but,  since  7  +  7  —  (2+1)=  11,  it  is  divisible  by  11. 
The  quotient  661  when  divided  by  6  gives  a  remainder  of  1,  which 
shows  that  it  may  be  a  prime  number.     It  cannot  be  divided  by  11, 
13,  17,  or  19,  and  is  seen  by  the  table  to  be  less  than  667  (23  X  29), 
and  not  equal  to  529  (232)  ;  therefore  it  is  a  prime  number. 
Therefore,  610,764  =  2a  X  3  X  7  X  11  X  661. 

Exercise  38. 

Find  the  prime  factors  of  : 

1.  148.  18.  179.  35.  65. 

2.  264.  19.  83.  36.  76. 

3.  178.  20.  2125.  37.  86. 

4.  183.  21.  2353.  38.  88. 

5.  173.  22.  2333.  39.  142. 

6.  187.  23.  165.  40.  326. 

7.  346.  24.  168.  41.  368. 

8.  343.  25.  2148.  42.  464. 

9.  210.  26.  16,662.  43.  292. 

10.  353.  27.  321.  44.  362. 

11.  5280.  28.  1551.  45.  365. 

12.  231.  29.  38.  46.  730. 

13.  31,416.  30.  82.  47.  42. 

14.  1369.  31.  129.  48.  868. 

15.  1368.  32.  72.  49.  999. 

16.  247.  33.  66.  50.  822. 

17.  327.  34.  68.  51.  1346. 


MEASURES   AND   MULTIPLES   OF   NUMBERS.  99 

52.  7641.  59.  128.  66.  78,309. 

53.  6234.  60.  8192.  67.  25,179. 

54.  234.  61.  8190.  68.  61,600. 

55.  579.  62.  8197.  69.  48,048. 

56.  577.  63.  3125.  70.  401,478. 

57.  212.  64.  2401.  71.  278,208. 

58.  126.  65.  1331.  72.  493,185. 

186.  A  number  is  not  only  divisible  by  each  of  its 
prime  factors,  but  by  every  possible  combination  of  them. 

Thus,  120  is  23  x  3  X  5,  and  is  divisible  by  2,  4,  8,  6,  12,  24,  30,  60, 
10,  20,  40,  or  15. 

187.  The  number  14.21  may  be  put  in  the  form  of 
1421  X  0.01  ;  and  thus  be  resolved  into  72  X  29  X  0.01. 
But  0.01  is  not  properly  a  factor,  it  is  a  divisor ;  it  is  the 
reciprocal  of  22  X  52.  Nevertheless,  it  is  frequently  of 
great  practical  advantage  to  separate  mixed  decimals  in 
this  way,  by  first  taking  out  the  apparent  factors,  0.1, 
0.01,  etc. 

Thus,  the  factors  of  142.1  may  be  said  to  be  7,  7,  29,  and  0.1 ;  of 
1.421,  7,  7,  29,  and  0.001. 


Exercise   39. 

Find  the 

prime  factors  of  : 

1.    8.4. 

9.   2.61. 

17. 

5.04. 

2.    7.6. 

10.    21.2. 

18. 

1.485. 

3.    1.08. 

11.    78.54. 

19. 

0.216. 

4.    0.144. 

12.  0.5236. 

20. 

34.87. 

5.    0.036. 

13.    0.00052. 

21. 

32.4. 

6.    0.037. 

14.    8.67. 

22. 

5.115. 

7.    21.45. 

15.    48.3. 

23. 

71.2. 

8.    14.6. 

16.   99.99. 

24. 

2.993. 

100      MEASURES   AND   MULTIPLES   OF   NUMBERS. 


Greatest  Common  Measure. 

188.  Measures  of  a  Number.  The  measures  of  a  num- 
ber are  the  exact  divisors  of  the  number. 

Thus,  a  man  with  five-dollar  bills  can  make  up  a  sum  of  $20,  but 
not  of  .$18.  A  wood  chopper  with  the  two-foot  mark  on  his  axe-handle 
can  measure  off  lengths  of  2,  4,  6,  or  8  feet,  but  not  of  3,  5,  or  7  feet. 
A  man  with  scales  and  a  four-ounce  weight  can  weigh  out  16  ounces 
of  tea,  but  not  22  ounces.  A  man  with  a  four-quart  measure  can 
measure  out  4,  8,  or  12  quarts  of  molasses,  but  not  5,  6,  or  7  quarts. 

189.  Common  Measures.  A  common  measure  of  two 
or  more  numbers  is  a  number  that  exactly  divides  each  of 
them. 

Thus,  $5  is  a  common  measure  of  $35  and  $40,  being  contained 
exactly  7  times  in  $35  and  8  times  in  $40.  3  feet  is  a  common  meas- 
ure of  21  feet,  15  feet,  and  12  feet.  1  yard  is  a  common  nuasure  of 
2  yards,  3  yards,  and  6  yards.  4  is  a  common  measure  of  12,  16, 
and  20. 

190.  Greatest  Common  Measure.  The  greatest  com- 
mon measure  of  two  or  more  numbers  is  the  greatest  num- 
ber that  exactly  divides  each  of  them. 

Thus,  the  measures  of  84  are  1,  2,  3,  4,  6,  7, 12,  14,  21,  28,  42,  and 
the  measures  of  36  are  1,  2,  3,  4,  6,  9,  12,  18. 

We  see  from  the  two  series  of  measures  that  1,  2, 3,  4,  6,  12  are  the 
only  measures  of  both  84  and  36,  and  that  12  is  the  greatest ;  there- 
fore, 12  is  the  greatest  common  measure  of  84  and  36. 

191.  The  letters  G.  C.  M.  stand  for  the  words  Greatest 
Common  Measure. 

Note.  Greatest  Common  Divisor  is  sometimes  used  instead  of  Great- 
est Common  Measure,  and  then  G.  C.  D.  is  used  instead  of  G.  C.  M. 

192.  If  two  integral  numbers  have  no  common  measure 
except  1,  they  are  said  to  be  prime  to  each  other. 

Thus,  27  and  32  are  prime  to  each  other,  though  both  are  compos- 
ite numbers. 


MEASURES   AND   MULTIPLES   OF   NUMBERS.       101 


193.   Examples.     1.  Find  the  G.  C.  M.  of  84,  126,  and 
210. 

Resolve  each  of  the  numbers  into  its  prime  factors. 


84 

2 

126 

42 

21 

7 

3 
3 

63 
21 

7 

7. 

126  =  2  X  32  X  7 

21210 


105 
_35 

7 


210  =  2  X  3  X  5  X  7. 


84  =  22  X  3  X  7 

The  factor  2  occurs  once  in  all  the  numbers. 
The  factor  3  occurs  once  in  all  the  numbers. 
The  factor  7  occurs  once  in  all  the  numbers. 
No  other  factor  occurs  in  all  the  numbeis. 
Therefore,  the  G.  C.  M.  is  2  X  3  X  7  =  42. 

2.   Find  the  G.  C.  M.  of  40  and  72. 

40  =  23  X  5.  72  =  23  X  32. 

Hence,  2  occurs  three  times  as  a  factor  in  40,  and  three  times  as  a 
factor  in  72.  No  other  factor  is  common  to  40  and  72.  Therefore, 
23,  or  8,  is  the  G.  C.  M.     Hence, 

194.  To  Find  the  G.  C.  M.  of  Two  or  More  Numbers, 

Separate  the  numbers  into  their  prime  factors.  Select 
the  lowest  power  of  each  factor  that  is  common  to  the  given 
numbers,  and  find  the  product  of  these  powers. 

195.  The  common  factors  of  two  or  more  numbers  may 
be  taken  out  at  the  same  time  as  follows  : 


2 

84 

126 

210 

3 

42 

63 

105 

7 

14 

21 

35 

2  3  5 

As  all  the  numbers  are  even,  2  is  a  common  factor.  As  3  is  an 
exact  divisor  of  42,  63,  and  105,  3  is  a  common  factor. 

As  7  is  an  exact  divisor  of  14,  21,  and  35,  7  is  a  common  factor. 

The  quotients  2,  3,  and  5  have  no  common  factor.  Therefore,  the 
only  common  factors  are  2,  3,  and  7  ;  and  the  G.  C.  M.  is  2  X  3  X  7  =  42. 


102      MEASURES   AND   MULTIPLES   OF   NUMBERS. 


Exercise  40. 

Find  the  G.  C. 

M.  of: 

1. 

27  and  33. 

7.  4,6,10. 

13.  96,  36,48. 

2. 

13  and  39. 

8.  9,  12,  21. 

14.  84,  105,  63. 

3. 

8  and  28. 

9.  10,  15,  25. 

15.  24,  60,84,128 

4. 

27  and  45. 

10.  14,  98,  42. 

16.  45,  81,  27,  90. 

5. 

81  and  108. 

11.  30,  18,  54. 

17.  78,  18,  54,42. 

6. 

4,  10,  12. 

12.  14,  56,42. 

18.  98,  28,70,  42. 

19.  96,  112: 

,  80,  32.     23.  252 

,  315,  420,  504. 

20.  24,  96, 

48,  120.    24.  128 

,  192,  320,  368,  432. 

21.  84,  252,  168,  210.       25.    136,  204,  357,  459. 

22.  33,  88,  77,  55.  26.    909,  1414,  2323,  4242. 

196.    Example.     Find  the  G.  C.  M.  of  69  and  184. 

Solution.     We  divide  the  greater  number 

69)184(2  by  the  smaller,  and  the  last  divisor  by  the  last 

138  remainder,  and  so  on  until  there   is  no  re- 

46)69(l  mainder.     The  final  divisor,  23,  is  the  greatest 

46  common  measure. 

23)46(2  This  method  of  finding  the  G.  C.  M.  can  be 

46    .  employed  when  the  numbers  cannot  readily 

be  separated  into  their  prime  factors. 

This  method  depends  upon  two  principles  : 

1 .  Every  factor  of  a  number  is  also  a  factor  of  every  multiple  of  that 
number. 

2.  Every  common  factor  of  two  numbers  is  also  a  factor  of  their 
sum  and  of  their  difference. 

Thus  4,  which  is  a  factor  of  12,  is  also  a  factor  of  24,  36,  etc.;  and 
6,  which  is  a  common  factor  of  24  and  36,  is  also  a  factor  of  60  and 
12. 

Let  us  apply  these  principles  to  this  example : 

Since  23  is  a  factor  of  itself  and  of  46,  it  is,  by  (2),  a  factor  of  69. 

Since  23  is  a  factor  of  69,  it  is,  by  (1),  a  factor  of  2  X  69,  or  138  ; 
and  therefore,  by  (2),  it  is  a  factor  of  138  +  46,  or  184. 

Hence,  23  is  a  common  factor  of  69  and  184. 


MEASURES    AND    MULTIPLES    OF    NUMBERS.       103 

Again,  every  common  factor  of  69  and  184  is,  by  (1),  a  factor  of 
2  X  69,  or  138  ;  and,  by  (2),  a  factor  of  184  —  138,  or  46. 

Every  such  factor,  being  now  a  common  factor  of  69  and  46,  is,  by 
(2),  a  factor  of  69  -  46,  or  23. 

Therefore,  the  greatest  common  factor  of  69  and  184  is  contained 
in  23,  and  cannot  be  greater  than  23.  And  23,  which  we  have  shown 
to  be  a  common  factor  of  69  and  184,  must  be  their  G.  C.  M. 

197.  In  the  course  of  this  operation  every  remainder 
contains,  as  a  factor  of  itself,  the  G.  C.  M.  sought ;  and 
this  G.  C.  M.  is  the  greatest  factor  common  to  that  re- 
mainder and  the  preceding  divisor.     Therefore, 

If  the  remainder  from  any  division  is  found  to  contain  a 
factor  that  is  not  a  factor  of  the  preceding  divisor,  the  re- 
mainder may  be  divided  by  that  factor,  and  the  quotient 
used  as  the  next  divisor. 

If  a  factor  common  to  any  remainder  and  the  preceding 
divisor  is  found,  both  remainder  and  divisor  may  be  divided 
by  the  common  factor,  and  that  factor  must  be  reserved  as  a 
factor  of  the  G.  C.  JH.  sought. 

198.  We  will  illustrate  by  two  examples. 

1.  Find  the  G.  C.  M.  of  4627  and  8593. 

4627)8593(1  t^  factor  6   is  thrown  out  of  the  first 

4627  remainder    3966,   for    it    is    prime    to    4627, 

6 1  3966  and  therefore  is  not  a  factor  of  the  G.  C.  M. 

661)4627(7  sought. 

4627  Therefore,  the  G.  C.  M.  is  661. 

2.  Find  the  G.  C.  M.  of  72,471  and  134,589. 

3172471            134589  The   common    factor  3  is  first 

24157           )  44863(l  taken  out  of  both  numbers.    From 

24157  the  remainder  20,706  the  factor  6, 

6 1 20706  which  is  prime  to  24,157,  is  ejected. 

345l)24157(7  The  ***  ^'  M'  is'  there^ore'  3  x 

24157         3451,  or  10,353. 


104      MEASURES    AND   MULTIPLES    OF   NUMBERS. 


Exercise  41. 

Find  the  G.  C.  M.  of  : 

1. 

2479  and  3589. 

11. 

44,323  and  61,087. 

2. 

3045  and  6195. 

12. 

232,353  and  39,699 

3. 

568  and  712. 

13. 

33,853  and  35,017. 

4. 

11,023  and  6493. 

14. 

5115  and  7254. 

5. 

1485  and  2160. 

15. 

2268  and  3348. 

6. 

7040  and  7392. 

16. 

1003  and  2419. 

7. 

2760  and  4485. 

17. 

419  and  52.301. 

8. 

1177  and  2675. 

18. 

30,072  and  133,784. 

9. 

78,473  and  94,653. 

19. 

4257  and  10.836. 

10. 

35,143  and  10,283. 

20. 

17,104  and  27,794. 

199.  To  find  the  G.  C.  M.  of  several  large  numbers,  we 
find  the  G.  C.  M.  of  two  of  the  numbers;  then  of  that 
result  and  a  third  number  ;  then  of  that  result  and  a 
fourth  ;  and  so  on.     The  last  G.  C.  M.  is  the  one  required. 

The  work  can  often  be  very  much  shortened  by  removing  from  each 
of  the  numbers  all  factors  less  than  13. 

Find  the  G.  C.  M.  of  3555,  4977,  and  6636. 


3 

8666 

4077 

6636 

Hence,  the  G.  C.  M. 

6 

1185 

3 | 1659 

4 | 2212 

required  is  3  X  79,  or 

3|237 

7J553 

7|553 

237. 

79 

79 

79 

Exercise  42. 

Fin 

d  the  G. 

C.  M.  of : 

1.  855,  1197,  1596.        5.  1177,  1391,  1819. 

2.  3864,  3404,  3657.        6.  4939,  1347,  3143. 

3.  15,561,  11.115,  13.585.    7.  740,  333,  296. 

4.  2943,  2616,  4578.        8.  833,  1785,  1309. 

9.  4994,  7491,  9988,  12,485,  16,571. 


MEASURES    AND    MULTIPLES    OF    NUMBERS.       105 

Least  Common  Multiple. 

200.  Multiples.  If  a  number  is  multiplied  by  an  in- 
teger, the  product  is  called  a  multiple  of  the  number. 

Thus,  $20  is  a  multiple  of  $5,  since  4  times  $5  is  $20. 

201.  A  series  of  multiples  of  a  number  is  found  by 
multiplying  the  number  by  the  integers,  1,  2,  3,  4,  5,  etc. 

Since  a  composite  number  is  the  product  of  only  one  set 
of  prime  numbers  (§  174),  every  multiple  of  a  number  con- 
tains all  the  prime  factors  of  the  number. 

202.  Common  Multiple.  A  multiple  of  two  or  more 
numbers  is  called  a  common  multiple  of  the  numbers. 

Thus,  6  X  $2  =  $12  ;  4  X  $3  =  $12  ;  3  X  $4  =  $12  ;  2  X  $6  =  $12. 
Therefore,  $12  is  a  common  multiple  of  $2,  $3,  $4,  and  $6. 

203.  Least  Common  Multiple.  The  smallest  common 
multiple  of  two  or  more  numbers  is  called  their  least  com- 
mon multiple  ;  and  it-  is  the  smallest  number  that  is  exactly 
divisible  by  each  of  them. 

Thus,  the  multiples  of  3  are  3,  6,  9,  12,  15,  18,  21,  24,  etc.;  and 
the  multiples  of  4  are  4,  8,  12,  16,  20,  24,  etc.  The  common  multiples 
of  3  and  4  are  12,  24,  etc. ;  and  the  smallest  of  these  is  12.  Therefore, 
the  least  common  multiple  of  3  and  4  is  12. 

204.  The  letters  L.  C.  M.  stand  for  the  words  Least 
Common  Multiple. 

205.  The  L.  C.  M.  of  two  or  more  numbers  is  a  number 
that  contains  all  the  prime  factors  of  each  of  these 
numbers.  Every  prime  factor,  therefore,  must  occur  in 
the  L.  C.  M.  the  greatest  number  of  times  it  occurs  as  a 
factor  in  any  one  of  them. 

Thus,  $20  =  2  X  2  X  5  X  $1,  and  $30  =  2  X  3  X  5  X  $1. 

The  L.  C.  M.  of  $20  and  $30  is,  therefore,  2  X  2  X  3  X  5  X  $1,  or  $60. 


106      MEASURES   AND   MULTIPLES   OF   NUMBERS. 


206.    Find  the  L.  C.  M.  of  84,  168,  252,  and  420. 

Solution.     Resolve  each  of  the  numbers  into  its  prime  factors. 

2 
2 
3 


84 

2 

168 

2 

252 

2 

420 

42 

2 

84 

2 

126 

2 

210 

21 

2 

42 

3 

63 

3 

106 

7 

3 

21 

7 

3 

21 

7  . 

5 

35 

7 

The  factor  2  occurs  three  times  in  168  ;  the  factor  3  occurs  twice  in 
262  ;  the  factor  6  occurs  once  in  420  ;  and  the  factor  7  occurs  once  in 
all  the  given  numbers. 

Therefore,  the  L.  C.  M.  is  28  X  32  X  5  X  7  =  2520.     Hence, 

207.  To  Find  the  L.  C.  M.  of  Two  or  More  Numbers, 
Separate  each  number  into  its  prime  factors.     Find  the 

product  of  these  factors,  taking  each  factor  the  greatest  num- 
ber of  times  it  occurs  in  any  one  of  the  given  numbers. 

208.  Examples.     1.  Find  the  L.  C.  M.  of  18,  24,  27,  45. 

Arrange  the  numbers  in  line,  and  divide  by  the  smallest  prime 
number  that  will  divide  two  or  more  of  the  numbers. 


2 

18 

24 

27 

46 

8 

9 

12 

27 

45 

:) 

4 

9 

15 

We  first  divide  by  2,  and  write  the  quotients  and  midivided  num- 
bers in  a  line  below.  In  the  first  line  of  quotients  we  cancel  9,  as  it 
is  an  exact  divisor  of  27,  and,  therefore,  27  contains  all  the  factors  of 
9.  We  next  divide  by  3,  and  the  quotients  by  3,  and  obtain  the  num- 
bers 4,  3,  6  in  the  last  line.  No  two  of  the  numbers  4, 3,  and  5  have  a 
common  factor.     Hence,  the  L.  C.  M.  is  2  X  3  X  3  X  4  X  3  X  5  =  1080. 

2.   Find  the  L.  C.  M.  of  3,  9,  27,  54. 

£,       ?,       2T,       54. 

We  cancel  the  3,  which  is  contained  in  9 ;  then  the  9,  which  is 
contained  in  27  ;  then  the  27,  which  is  contained  in  54,  and  have  54 
for  the  L.  C  M.  of  the  numbers. 


MEASURES   AND   MULTIPLES   OF   NUMBERS.       107 

3.   Find  the  L.  C.  M.  of  13,  15,  26,  39  : 

S)X?  15  26  39 


26 


n 


We  cancel  the  13  of  the  first  line  and  divide  by  3,  getting  5,  26,  13. 
We  cancel  the  13  of  this  line.     The  L.  C.  M.  is  3  X  6  X  26  =  390. 


Exercise  43. 

Find  the  L.  C.  M.  of  : 

1. 

6,  14,  21. 

26. 

30,  42,  105,  70. 

2. 

8,  12,  3,  24. 

27. 

36,  24,  35,  20. 

3. 

6,  10,  15. 

28. 

7,  11,  14,  15. 

4. 

9,  12,  18,  4. 

29. 

12,  18,  27,  63,  28. 

5. 

15,  21,  35. 

30. 

34,  26,  65,  85,  51,  39. 

6. 

12,  20,  24. 

31. 

12,  18,  96,  144. 

7. 

14,  24,  28. 

32. 

84,  156,  63,  99. 

8. 

12,  15,  20. 

33. 

17,  51,  119,  210. 

9. 

16,  24,  32. 

34. 

16,  30,  48,  56,  72. 

10. 

21,  33,  77. 

— 

35. 

27,  33,  54,  69,  132. 

11. 

27,  33,  99. 

36. 

15,  26,  39,  65,  180. 

12. 

7,  11,  13. 

37. 

44,  126,  198,  280,  330, 

13. 

77,  55,  35. 

38. 

50,  338,  675,  975. 

14. 

16,  18,  27,  72. 

39. 

552,  575,  920. 

15. 

10,  12,  22,  33, 

60. 

40. 

228,  304,  342. 

16. 

15,  16,  18,  20, 

22,  24. 

41. 

1080  and  1260. 

17. 

56,  64,  70,  84, 

112. 

42. 

600  and  480. 

18. 

48,  54,  81,  144 

:,  162. 

43. 

1564  and  1932. 

19. 

75,  100,  120,  150,  180. 

44. 

2530  and  1760. 

20. 

112,  168,  196, 

224. 

45. 

936  and  2925. 

21. 

7,  14,  15,  21,  45. 

46. 

3432  and  4032. 

22. 

16,  25,  81. 

47. 

1875  and  2425. 

23. 

26,  39,  52,  65. 

48. 

1632  and  2976. 

24. 

80,  72,  225,  48 

49. 

1001  and  2233. 

25. 

10,  20,  30,  40, 

50,  60. 

50. 

539  and  1463. 

108      MEASURES   AND   MULTIPLES   OF   NUMBERS. 

209.  If  the  given  numbers  are  large  and  contain  no 
prime  factors  that  can  readily  be  detected,  it  is  best  to 
obtain  the  common  factors  by  the  process  for  finding  the 
G.  C.  M.  under  like  circumstances. 

Example.     Find  the  L.  C.  M.  of  1247  and  1769. 


1247)1769(1 
1217 

2j 
9 1 


622 


261 
29)1247(43 
116 
87 
87 


Hence,  the  G.  C.  M.  of  1247  and  1769  is  29  ; 
and  1247  =  29  x  43, 
and  1769=29  X  <il. 
Therefore,  the  L.  C.  M.  of  1247  and  1769 
is  29  X  43  X  61  =  1247  X  61,    or   1769  X  43, 
that  is,  76,067. 


210.  From  this  process  it  will  be  seen  that : 

The  L.  C.  M.  of  two  numbers  may  be  found  by  dividing 
either  of  the  numbers  by  their  G.  C.  M.  and  multiplying  the 
quotient  by  the  other  number. 

211.  The  L.  C.  M.  of  two  prime  numbers,  or  of  two 
numbers  prime  to  each  other,  is  their  product. 


Exercise  44. 

Find  the  L.  C.  M. 

of: 

1. 

424  and  583. 

11. 

3864,  3404,  3657. 

2 

319  and  407. 

12. 

539  and  253. 

3. 

1679  and  1932. 

13. 

2943,  2616,  4578. 

4. 

1003  and  2419. 

14. 

2863  and  1151. 

5. 

1003  and  1357. 

15. 

1177,  1391,1819.. 

6. 

899  and  961. 

16. 

5317  and  2863. 

7. 

407,  703,  444. 

17. 

12,703  and  12,879. 

8. 

411,  959,  2055. 

18. 

23,309  and  10,753. 

9. 

221  and  351. 

19. 

4939  and  3143. 

10. 

1426  and  989. 

20. 

4199  and  6137. 

CHAPTEE   VII. 


COMMON  FEACTIONS. 

212.    What  is  the  name  of  one  of  the  parts  when  a  unit 
is  divided  into  : 


1. 

2. 
3. 
4. 
5. 

Two  equal  parts  ?              6.    Eight  equal  parts  ? 
Three  equal  parts  ?            7.    Ten  equal  parts  ? 
Four  equal  parts  ?             8.    Twelve  equal  parts? 
Five  equal  parts  ?              9.    Twenty  equal  parts  ? 
Six  equal  parts  ?              10.    One  hundred  equal  parts  ? 

213.    A  unit  contains  how  many  : 

1. 

Halves?            7.   Ninths?             13.    Twentieths? 

2. 

Thirds?            8.    Sevenths?          14.    Twenty-fourths? 

3. 

Fourths?          9r  Tenths?             15.    Thirtieths? 

4. 

Fifths?           10.    Twelfths?          16.    Fortieths? 

5. 

Sixths?           11.    Elevenths?        17.    Fiftieths? 

6.    Eighths?        12.    Fifteenths?        18.    Hundredths? 

214.   When  a  unit  is  divided  into  twelve  equal  parts, 
what  is  the  name  of  : 

1;    One  part? 

2.  Three  parts  ? 

3.  Six  parts  ? 


4.  Two  parts  ? 

5.  Five  parts  ? 

6.  Four  parts  ? 


7.  Eight  parts  ? 

8.  Ten  parts  ? 

9.  Twelve  parts  ? 


215.  Equal  parts  of  a  unit  are  called  fractional  parts  of 
the  unit. 

216.  In  three  quarters  of  a  yard,  the  unit  counted  is  a 
quarter  of  a  yard. 


110  COMMON    FRACTIONS. 

217.  A  unit  that  is  a  fractional  part  of  a  whole  unit  is 
called  a,  fractional  unit. 

218.  Fractions.  Numbers  that  count  fractional  units 
are  called  fractions. 

219.  Common  Fractions.  A  fraction  expressed  by  two 
numbers  one  under  the  other  with  a  line  between  them 
is  called  a  common  fraction. 

220.  Simple  Fractions.  If  the  two  numbers  are  whole 
numbers,  the  fraction  is  called  a  simple  fraction. 

Thus,  three  fifths,  written  \ ,  is  a  simple  fraction. 

221.  The  lower  number  is  called  the  denominator  (name- 
giver),  the  upper  number  is  called  the  numerator  (number- 
giver)  ;  and  the  numerator  and  denominator  together  are 
called  the  terms  of  the  fraction. 

222.  The  fraction  $  means  3  times  J,  where  £  is  the 
fractional  unit  and  3  is  the  number  of  them. 

223.  The  fractional  unit  is  expressed  by  1  divided  by  the 
denominator,  and  the  number  of  fractional  units  taken  is 
expressed  by  the  numerator. 

Name  the  fractional  unit  and  the  integral  unit  of : 

1.  f  of  an  inch.  5.  £  of  a  week. 

2.  $  of  a  dollar.  6.  T75  of  a  foot. 

3.  |  of  a  bushel.  7.  ^  of  a  pound. 

4.  §  of  a  yard.  8.  f  of  an  acre. 

224.  To  Read  a  Common  Fraction, 

Read  the  numerator  and  thert  the  denominator. 

Thus,  £,  |,  f ,  f ,  \,  T8T  are  read  one  half,  two  thirds,  three  fourths 
or  three  quarters,  two  fifths,  one  sixth,  eight  elevenths. 


COMMON    FRACTIONS.  Ill 

Bead:     |j  |;  |j  Vs  A*  if;  Ai  Ari  Hi  AV 

Express  in  figures  : 

1.  Two  thirds.  5.  Eleven  sixteenths. 

2.  Five  sevenths.  6.  Seventeen  twentieths. 

3.  Seven  ninths.  7.  One  twenty-fifth. 

4.  Eight  tenths.  8.  Thirty  hundredths. 

225.  A  proper  fraction  is  a  fraction  whose  numerator 
is  less  than  its  denominator  ;  as  J. 

226.  An  improper  fraction  is  a  fraction  whose  numerator 
is  not  less  than  its  denominator  ;  as  §,  J^7-. 

Note.  When  the  numerator  is  greater  than  the  denominator, 
more  than  one  unit  must  be  regarded  as  divided  into  equal  parts. 

Thus,  £  means  that  three  units  have  been  divided  each  into  four 
equal  parts,  and  that  all  the  parts  of  two  units  and  one  part  of  the 
third  unit  are  taken  ;  or,  £  means  that  nine  units,  considered  as  one 
quantity,  have  been  divided  into  four  equal  parts,  and  that  one  of 
these  parts  has  been  taken. 

227.  How  many  integral  units  must  be  divided  into 
equal  parts  that  we  may  have  f  of  the  unit  ?  §  ?  y5  ? 
£  ?  2£  ?  y  ?  ^7  ?  yt  ? 

228.  A  mixed  number  is  a  whole  number  and  a  frac- 
tion ;  as  4f ,  5.35.  These  are  read  four  and  three  sevenths, 
five  and  thirty-five  hundredths. 

Note.  Every  mixed  number  means  that  some  entire  units  are 
taken  and  the  fraction  of  another  unit. 

229.  Name  the  proper  fractions,  the  improper  fractions, 
and  the  mixed  numbers  of  the  following  expressions  : 

3*1  § ;  *&\  I;  tV;  ft;  u»;  3«;  Ty^;  v;  if;  \%h 

18H;  «;  27^;  fj;   V1-;  «;  3*;  T\;  V- 


112  COMMON   FRACTIONS. 


Reduction  of  an  Improper  Fraction  to  a  Whole  or  a  Mixed 

Number. 

230.  An  improper  fraction  represents  a  quantity  which 
can  also  be  represented  by  a  whole  number,  or  else  by  a 
mixed  number. 

Thus,  ^=  2f;  for,  if  we  suppose  several  units  to  be  divided  each 
into  seven  equal  parts,  and  we  take  17  of  these  parts,  14  (that  is,  2X7) 
will  make  two  units,  and  the  three  parts  remaining  will  be  three 
sevenths  of  another  unit.     Hence, 

231.  To  Reduce  an  Improper  Fraction  to  a  Whole  or  a 
Mixed  Number, 

Divide  the  numerator  by  the  denominator. 

The  quotient  will  be  the  whole  number,  and  the  remainder,  if  any, 
will  be  the  numerator  of  the  fractional  part,  of  which  the  denominator 
will  be  the  denominator  of  the  improper  fraction. 

Exercise  45. 
Reduce  to  a  whole  or  a  mixed  number  : 

1.    if..  8.    V-  »•    V-  22-    W- 


2. 

¥• 

9. 

1 

16. 

¥• 

23. 

w- 

3. 

¥• 

10. 

¥• 

17. 

* 

24. 

w- 

4. 

W- 

11. 

«• 

18. 

w- 

25. 

w- 

5. 

V* 

12. 

ft. 

19. 

w- 

26. 

AIP. 

6. 

V 

13. 

«. 

20. 

w- 

27. 

m*- 

7. 

¥« 

14. 

«• 

21. 

w- 

28. 

¥t¥« 

Reduction  of  a  "Whole  or  a  Mixed  Number  to  an  Improper 
Fraction. 

232.    A  whole  number  may  be  expressed  as  a  fraction 
with  any  given  denominator. 

Thus,  7  =  A,8- ;  for,  as  each  unit  contains  9  ninths,  7  units  contain 
7X9  ninths,  that  is,  63  ninths.     Hence, 


COMMON    FR ACTIONS.  113 

233.  To  Reduce  a  Whole  Number  to  a  Fraction, 

Multiply  the  whole  number  by  the  denominator  of  the  required 
fraction,  and  under  this  product  write  the  denominator. 

Exercise  46. 

Reduce  to  an  improper  fraction  : 


1. 

4  =  -*-. 

5. 

U=  r- 

9. 

9  =  „. 

2. 

5  =  -T-. 

6. 

7  =  -7-. 

10. 

18  =  TT. 

3. 

«  =  -»- 

7. 

3=5- 

11. 

12  =  „. 

4. 

8=*- 

8. 

14  =  T*- 

12. 

16=„. 

234.  A  mixed  number  represents  a  quantity  that  can 
also  be  represented  by  an  improper  fraction. 

Thus,  6/T  =  || ;  for  each  unit  contains  13  thirteenths ;  therefore  5 
units  contain  6  X  13  thirteenths,  or  65  thirteenths;  which,  together 
with  the  7  thirteenths,  make  72  thirteenths.     Hence, 

235.  To  Reduce  a  Mixed  Number  to  a  Fraction, 

Multiply  the  whole  number  by  the  denominator  of  the  frac- 
tion, and  to  the  product  add  the  numerator  ;  under  this  sum 
write  the  denominator. 

Exercise  47. 
Reduce  to  an  improper  fraction  : 

1.  3|.  10.  44f.  19.  14}$.  28.  36f£. 

2.  5TV  11.  2^|.  20.  21^V  29.  11^. 

3.  12TV  12.  3||.  21.  6U.  30.  3H- 

4.  8f  13.  10TV  22.  16^.  31.  16H- 

5.  25f.  14.  121-f.  23.  lljf.  32.  15?V 

6.  17 J.  15.  84^.  24.  8TV  33.  108^. 

7.  8 A-  16-  16£f.  25.  12f.  34.  51TV 

8.  9^.  17.  17TV  26.  27£.  35.  40}f 

9.  162T\.  18.  19f.  27.  111$.  36.  864^. 


114  COMMON    FRACTIONS. 

Reduction  of  a  Fraction  to  Lower  Terms. 

a\     I     1     1     I     I     1     1     I     1     1     1     I     1     I     1* 
C  D  E  F 

236.  If  AB  is  divided  into  5  equal  parts,  how  many  of 
these  parts  are  there  in  AC ?     What  part  of  AB  is  AC? 

If  AB  is  divided  into  15  equal  parts,  how  many  of  these 
parts  are  there  in  AC?     What  part  of  AB  is  AC? 
Which  is  the  greater  fraction,  {§  or  $  ? 
What  must  be  done  to  \$  to  make  f  ? 
Which  is  greater,  \  or  £  ?  ^  or  \  ?  \\  or  f  ?     Hence, 

237.  To  Reduce  a  Fraction  to  Lower  Terms, 

Divide  both  numerator  and  denominator  by  any  common 
factor. 

238.  A  fraction  is  expressed  in  its  lowest  terms  when 
the  numerator  and  denominator  are  prime  to  each  other. 

239.  Examples.     1 .   Reduce  $££  to  its  lowest  terms. 

Ht=lf=AsstJ 

the  common  divisors  used  being  9,  4,  and  3. 

2.  Reduce  §£§  to  its  lowest  terms. 

We  first  find  the  prime  factors  of  333  to  be  3,  3,  and  37. 
Since  the  factor  3  of  333  will  not  exactly  divide  259  we  try  37,  and 
find  it  is  contained  7  times  in  259. 

Dividing  259  and  333  each  by  37,  we  have  f f|  =  $. 

3.  Reduce  Iff  £  to  lts  lowest  terms. 

Since  no  common  factor  can  be  readily  detected,  we  find  the  G.  C.  M. 
of  1261  and  1649  to  be  97. 

Dividing  1261  and  1649  each  by  97,  we  have  \\\\  =  jf     Hence, 

240.  To  Reduce  a  Fraction  to  its  Lowest  Terms, 
Cancel  all  the  factors  common  to  the  numerator  and  denomi- 
nator ;  or  divide  both  terms  by  their  G.  C.  M. 


COMMON   FRACTIONS.  115 

V 

Exercise  48. 
Reduce  to  lowest  terms  : 
i.    B*.  8-    t¥AV  15.    m%.         22.    iffff. 


2. 

«f 

3. 

T322~V 

4. 

Hit- 

5. 

Htf- 

6. 

MIS- 

7. 

iltl- 

9. 

MM- 

16. 

T§Mtf- 

23. 

*»lf 

10. 

tVsV 

17. 

Hit- 

24. 

tM&- 

11. 

MH- 

18. 

T§M' 

25. 

Ht£- 

12. 

tVsV 

19. 

51  6 
STOT' 

26. 

Hill- 

13. 

•HI- 

20. 

3  8  7  2 
¥2"F(TT* 

27. 

««*• 

14. 

*WA- 

21. 

7  8  4  7  3 

28. 

mm 

241.  In  practice,  in  the  answers  to  all  examples,  unless 
the  problem  expressly  demands  the  contrary,  every  frac- 
tion is  left  in  its  lowest  terms,  and  every  improper  fraction 
is  reduced  to  a  whole  or  a  mixed  number. 

Reduction  of  a  Fraction  to  Higher  Terms. 

242.  How  many  quarters  are  there  in  -£-?  Which  is 
greater,  -J-  of  an  apple  or  f  of  an  apple  ? 

How  many  sixths  of  an  orange  can  be  cut  from  -J  of  an 
orange  ?  -J-  of  an  orange  ?  f  of  an  orange  ? 

Which  is  greater,  £  or  T\  ?  f  0r  Te¥  ?  f  or  £|  ?  }  or  §§  ? 
By  what  must  we  multiply  both  terms  to  change  T6T  to 

H*  ftoM?  *to«? 

243.  To  Reduce  a  Fraction  to  Higher  Terms, 

Divide  the  required  denominator  by  the  denominator  of  the 
given  fraction,  and  multiply  both  terms  of  the  fraction  by  the 
quotient. 

Exercise  49. 


Reduce : 

1.   fto20ths. 

4. 

T7s  to  39ths. 

7. 

T\  to  144ths. 

2.    f  to  24ths. 

5. 

T\  to  90ths. 

8. 

T7¥  to  144ths. 

3.    §  to  50ths. 

6. 

§  to  108ths. 

9. 

TV  to  156ths. 

116  COMMON   FRACTIONS. 

Multiplication  of  Fractions. 

244.  Compound  Fractions.  A  fraction  of  a  whole  num- 
ber, of  a  mixed  number,  or  of  a  fraction,  is  called  a  com- 
pound fraction. 

245.  The  expression  |  X  i  means  the  same  as  $  of  £, 
and  the  sign  X  should  be  read  of  or  multiplied  by  when  it 
follows  a  fraction. 

246.  7  times  3  horses  are  how  many  horses  ? 

7  times  three  fifths  ($ )  are  how  many  fifths  ? 
£  of  12  men  are  how  many  men  ? 
12  X  $£  are  how  many  dollars  ? 

247.  To  Find  the  Product  of  a  Whole  Number  and  a 
Fraction, 

Divide  the  product  of  the  numerator  and  the  whole  number 
by  the  denominator. 

Note.  A  factor  common  to  the  whole  number  and  the  denominator 
should  be  cancelled.     Thus, 

o       8 

Exercise  50. 
Find  the  product  of  : 

1.  £X2.       10.  £$X2.  19.  ££Xl5.  28.  Jf  X  90. 

2.  |X9.       11.  f$X3.  20.  £§X20.  29.  f  of  434. 

3.  10  X£.     12.  ^|X4.  21.  ,&of324.  30.  468  X  y. 

4.  15  X§.     13.  5  X}}.  22.  ^  of  273.  31.  30  X  If 

5.  3&X7.     14.  6X^|.  23.  |T  of  242.  32.  100  X  |f . 

6.  16  X  j.     15.  7  X  ft  24.  340  X  TV  33.  ?$  X  54. 

7.  |X2.       16.  8  X^.  25.  450  X^.  34.  §£  X  48. 

8.  fVX5\     17.  ^X10.  26.  T^X1000.  35.  72  X  |$. 

9.  27  Xf     18.  JgXl2.  27.  &  X  210.  36.  Jj  of  128. 


COMMON    FRACTIONS.  117 

248.   To  Multiply  a  Fraction  by  a  Fraction. 
Multiply  f  by  |. 


M    \    1    I    I    I    I    I    I    I    I    I    I    1    1    \B 
C  D  E  F 

To  multiply  f  by  §  means  to  find  f  of  i. 

If  the  line  AB  is  divided  into  5  equal  parts  at  the  points  C,  D,  E, 
and  F,  AF  will  be  4  of  these  parts,  or  4  of  AB. 

If  each  part  is  subdivided  into  three  equal  parts,  there  will  be  15 
of  these  smaller  parts  in  the  whole  line,  and  each  part  will  be  y1^  of  the 
line.     Therefore,  |  of  A  is  Tas  of  the  whole  line. 

Now  |  of  f  is  4  times  as  much  as  }  of  |,  or  r45  of  the  whole  line. 

And  f  of  f  is  twice  as  much  as  £  of  f ,  or  T85  of  the  whole  line. 

249.   Therefore,  we  have  the  following 

Kule.  Find  the  product  of  the  numerators  for  the  required 
numerator,  and  the  product  of  the  denominators  for  the 
required  denominator. 

Note.  Mixed  numbers  and  whole  numbers  are  brought  under  this 
rule  by  first  reducing  them  to  improper  fractions. 

Any  factor  common  to  a  numerator  and  a  denominator  should  be 
cancelled  before  multiplying. 

Example.    Find  the  product  of  Tf  X  2T%  X  {%. 

2       2 

3 

We  first  reduce  the  mixed  number  2-fc  to  an  improper  fraction  and 
obtain  f  f . 

We  cancel  the  13  of  the  numerators  and  the  13  of  the  denominators. 
We  cancel  the  common  factor  17  of  the  34  and  the  17,  and  the 
common  factor  5  of  the  10  and  the  15. 

There  remains  in  the  numerators  2X2,  and  in  the  denominators  3, 
from  which  we  obtain  the  improper  fraction  |. 


118  COMMON   FRACTIONS. 

Exercise  51. 
Find  the  product  of  : 

1.  ioift.  10-  *X4f 

2.  $  of  2^.  11.  fofAof^ofiof  JoflSj. 

3.  }  off  12.  5JX8|. 

4.  2}X2f  13.  fX|XAX7f 

5.  4f  X2f  14.  }of«of  Aof8f 

6.  4f  X9f  15.   A-XHXJIX2H. 

7.  i  of  J  of  10.  16.  JJX^XM,. 

8.  J  of  J  of  $.  17.  |  X  Iff  X  ff  X  17. 

9.  *  X  |  X  |  X  4f  18.   3|  X  «  X  H  X  1||. 

19.  i  of  f  of  f  of  #  of  f  of  |  of  |  of  A  of  10. 

20-  A  0^  A  of  30.  36.  37*  X  H  X  M  X  Jf 

21.  HI  X  ,flfr  X  H  X  If.  87.  A  X  ft  X  A  X  2f 

22.  |XiXAXfof|ofiof8.  38.  8*  X  ft  X  1^  X  f 

23.  A  of  |»  of  rfV  39.  62^  X  A  X  |  X 15. 

24.  ^T  X  TV  X  | §  X  48.  40.   ft  X  87*  X  ft  X  f 

25.  ||  of  ^  of  |f  of  12.  41.  1|X1^X  3ft  X  ft. 

26.  If-  X  4*  X  |.  42.  6|  X  H  X  |  X  |. 

27.  2|  X  If  X  1«  X  8.  43.  ft  of  ft  of  ||  of  lOf 

28.  3f  X  2i  of  1 A  X  1  A-  44.  ||  X  2J|  X  If  X  27. 

29.  H  X  5f  X  4i  X  ft  X5.  45.  2^  X  Iff  X  TJV  X  2^. 

30.  fofAX8»XAoflH-*«-  H  X  l^fr  X  |J  X  12|. 

31.  H  X  ||  X  1||.  47.  HI  X  lft  X  ||  X  ft. 
32-  |f|  X  «f  X  tfft.  48.  3|  X  23*ff  X  lft  X  J|. 

33.  IflJof  AWofWI-         49-   HiX||XAXlA. 

34.  A  X  n  X  6|  X  |f  50.  15f  X  ft  X\%X  f  f 

35.  12J  X  ft  X  16»  X  A-         ■«•  m  X  V  X  A  X  Jf 

52.  lrfTXl|jXi||XHf 

53.  f  X  ^T  X  6f  X  9|  X  2*  X  63  X  jft. 

54.  6£  X  11 J  X  16T\  X  ft  of  ft  of  A- 

55.  2|  X  7ft  X  2  X  1J  X  A  X  577  X  |f 


COMMON   FRACTIONS. 


119 


250.  To  Find  the  Product  of  a  Mixed  Number  and  a 
Whole  Number, 

Find  the  product  of  the  whole  number  and  the  fractional 
part  of  the  mixed  number,  then  the  product  of  the  whole 
number  and  the  integral  part,  and  add  these  products. 


1.    Multiply  7£  by  9. 

n 

9 
64J 

In  Ex.  1,  9  times  i  equals  f,  or  1|. 
1  to  the  product  of  9  X  7,  making  64. 

In  Ex.  2,  |  of  8  equals  J£,  or  5] 
5  to  the  product  of  2  X  8,  making  21. 


2.    Multiply  8  by  2§. 
8 

n 

We  write  the  |,  and  add  the 
,    We  write  the  i,  and  add  the 


Exercise  52. 


Find  the  product  of  : 


1.  9X6|. 

2.  8  X  17i. 

3.  19  X  5J. 

4.  7X12J. 

5.  10X154. 

6.  6Xlf 

7.  12X2|. 

8.  17X6£. 

9.  19XlTV- 

10.  24  X  16f 

11.  32X22§. 

12.  40X8£. 

13.  41x9£. 

14.  18  X  7f . 

15.  19X6^. 

16.  20X5£. 


17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 


15  X  $9*. 
6X8}. 
HX8f. 
100  X  6§. 
5X34. 
6  X  17f 
32  X  6f. 
13  X  3f 
12  X  6} . 
8T2T  X  12. 
20i  X  5. 
6§  X  18. 

11  x  114. 
18  X  12$. 
36  X  44. 

12  X  20f . 


33.  12X48||. 

34.  11X24|. 

35.  7  Xl9f. 

36.  8  X  16J. 

37.  5  X  29|, 

38.  16  X3f 

39.  19X12^. 

40.  23  X  42f . 

41.  18  X  I24. 

42.  22  X  22,V 

43.  12  X  161}}. 

44.  9X144*. 

45.  10  X  II24. 

46.  14X42|. 

47.  161  X  4§. 

48.  140  X  5TV 


120 


COMMON    FRACTIONS. 


Division  of  Fractions. 

251.  The  reciprocal  of  a  fraction  is  the  fraction  with  its 
terms  interchanged. 

Thus,  the  reciprocal  of  f  is  J,  for  J  X  f  =  1. 

To  write  the  reciprocal  of  a  whole  or  a  mixed  number,  we 
reduce  the  number  to  an  improper  fraction,  and  write  the 
improper  fraction  with  its  terms  interchanged. 

Thus,  the  reciprocal  of  4  is  i;  the  reciprocal  of  2|  is  f. 

Multiplying  by  the  reciprocal  of  a  number  gives  the  same 
result  as  dividing  by  the  number  (§101).     Hence, 

252.  To  Divide  by  a  Whole  Number  or  a  Fraction, 

Multiply  the  dividend  by  the  reciprocal  of  the  divisor. 


Exercise  53. 

Divide  : 

1.  §£by6. 

15.  5  by  4f. 

29.  lt|  by  f . 

2.  H  by  5. 

16.  4§  by  }. 

30.  100  by  83£. 

3.  9  by  8. 

17.  8}  by  6f 

31.  50  by  16§. 

4.  18$  by  7. 

18.  St  by  1^. 

32.  H  by  If 

5-  |  by  f. 

19.  100  by  6$. 

33.  liibylA- 

6-  «  by  f 

20.  H  by  If. 

34.  20±  by  5. 

7.  If  by  3f 

21.  3£by  5. 

35.  16f  by  |. 

8.  5*  by  4§. 

22.  100  by  33£. 

36.  22f  by  16jj. 

9.  8}  by  4f 

23.  100  by  37£. 

37.  20f  by  ltf. 

10.  7Jby4f. 

24.  7}  by  6f 

38.  16$  by  111. 

11.  6f  by9f 

25.  J  by  TV 

39.  33J  by  28| . 

12.  8§by4§. 

26.  6f  by  32. 

40.  471  by  171- 

13.  3jby«. 

27.  3}  by  3f 

41.  18*byl5V 

14.  4$  by  6f. 

28.  1A  by  1J. 

42.  37$  by  1^. 

43.  3J  of  2±  by  1£ 

of  2J.      45.  2T\  ( 

>f  51  by  7|. 

44.  2f  by  3iof  1TV              46.  5$  of 

81ofl^by2TVof5|. 

COMMON    FRACTIONS.  121 


A  Short  Method  of  Dividing  a  Mixed  Number  by  a 
Whole  Number. 

253.     1 


Divide  16£  by  4. 

2.   Divide  16§  by  7. 

4)_16| 

7)18! 

H 

2/t 

In  the  first  problem  we  simply  divide  the  whole  number  16  by  4, 
and  then  the  fraction  |  by  4,  and  obtain  the  result  at  once,  4i. 

In  the  second  problem  we  divide  the  16  by  7,  and  obtain  the 
quotient  2  and  a  remainder  2.  The  remainder  2  is  joined  with  the  f , 
making  2|,  or  f ,  and  §  —•  7  =  ^8T. 

This  method  is  the  shortest  method  of  dividing  a  mixed  number  by 
a  whole  number. 

Exercise  54. 
Find  the  quotient  of  : 

1.  31£-^5.  5.  42f-^6.  9.  48||4-12. 

2.  16^^-6.  6.  49J-T-7.  10.  24f-f-ll. 

3.  14f^-2.  7.  52f-r-8.  11.  19f~7. 

4.  33£-h7.  r  8.  44T\4-12.  12.  29J-f-8. 

254.    Examples.     1.   Find  the  value  of  (2£  «rf)  X  f. 

3 

(2 1  t  f)  x  f  =  2  x  g  x  I  —  I  ~  li- 

We  have  to  divide  2£  by  { ,  and  to  multiply  the  result  by  f.  But 
2^  is  divided  by  §  by  multiplying  2$  by  f.  Hence,  we  invert  the 
divisor  £  and  find  the  product  of  the  three  fractions. 

2.   Find  the  value  of  2£-f-  f  of  f. 

We  have  to  divide  2\  by  |  of  f  ;  hence  we  may  find  the  product  of 
f  and  |,  and  invert  this  product,  or  we  may  invert  both  factors  of  this 
product  and  multiply  by  the  inverted  factors  f  and  f . 


122  COMMON    FRACTIONS. 

Exercise  55. 
Find  the  value  of  : 

1.  2jof2*-r-A<>f3|.         9.  |  of  lA-r-Hofff 

2.  |of6§of  A"^*.         10.  »  of  |J-=- A  of  4. 

3.  A -^1  of  2*  of  If         11.  A  of  Hpr^|oflTV 

4-  A^-(*X2iXlf).       12.  fofffof  A-KiXJoft). 

5.  Jofil-rlAoflH.      13.   |off  ofif-rjof  AoflA- 

6.  |of|-=-(|XA).         14-  tt-s-H)-^(6AH-4«). 

7.  I  of  H-T-Hofjf.         15.  (14|-r4|)  +  (3H-^9|). 

8-  fofff-rHof  if         16-  fof  JJof8i-r-3Aof  Aofoi. 

255.  Example.  If  f  of  a  barrel  of  flour  costs  $3, 
what  is  the  cost  of  ±  of  a  barrel  ?  What  is  the  cost  of  a 
barrel  ? 

Solution.  Since  £  of  a  barrel  of  flour  costs  $3,  \  of  a  barrel  will 
cost  \  of  $3,  or  $1.  If  £  of  a  barrel  costs  $1,  a  barrel  will  cost  4  X  $1, 
or  $4.     Hence, 

256.  To  Find  the  Whole  when  a  Fractional  Part  is 
Given, 

Divide  the  given  part  by  the  numerator  of  the  fraction  and 
multiply  the  quotient  by  the  denominator. 

Exercise  56. 

1.  If  |  of  a  ton  of  hay  costs  $15,  what  is  the  cost  of 
one  ton  ? 

2.  15  is  |  of  what  number  ? 

3.  If  $  of  a  roll  of  carpeting  is  worth  $75,  what  is  the 
whole  roll  worth  ? 

4.  A  man  sold  6f  yards  of  cloth,  which  was  A  of  tne 
whole  piece.     How  many  yards  were  there  in  the  piece  ? 

5.  A  farmer  sold  $  of  his  hay  for  $195.60.    What  was 
the  value  of  his  entire  crop  of  hay  ? 


COMMON    FRACTIONS.  123 

6.  21}  is  jf  of  what  number  ? 

7.  6f  is  ^  of  what  number  ? 

8.  2^|  is  |f  of  what  number  ? 

9.  If  f  of  an  acre  of  land  is  worth  $32,  what  is  the 
value  of  an  acre  ? 

10.  If  f  of  a  bushel  of  wheat  is  worth  48  cents,  what  is 
the  value  of  2T7^  bushels  of  wheat  ? 

11.  If  f  of  a  ton  of  hay  is  worth  $15,  what  is  the  value 
of  7£  tons  of  hay  ? 

12.  If  |  of  a  cord  of  wood  is  worth  $4,  find  the  value 
of  7  cords  of  wood  ? 

13.  If  T4T  of  a  barrel  of  apples  is  worth  44  cents,  what 
is  the  value  of  12  barrels  of  apples  ? 

14.  $125  is  i  more  than  (that  is,  |  of)  what  sum  of 
money  ? 

15.  $132  is  £  less  than  what  sum  of  money  ? 

16.  495  is  -J-  more  than  what  number  ? 

17.  217  is  -J-  less  than  what  number  ? 

18.  495  is  T2jj  less  than  what  number  ? 

19.  495  is  -j^-  more  than  what  number  ? 

20.  If  |  of  a  yard  of  silk  is  worth  $1,  find  the  value  of 
4  yards  of  silk. 

21.  If  |  of  a  yard  of  linen  is  worth  60  cents,  what  is 
the  value  of  2£  yards  of  linen  ? 

22.  If  a  man  who  owned  -J-  of  a  schooner  sold  f  of  his 
share  for  $1200,  what  was  the  value  of  the  schooner  ? 

23.  One  fourth  of  one  third  of  three  sevenths  of  a  num- 
ber is  60.     What  is  the  number  ? 

24.  Three  fourths  of  two  ninths  of  six  sevenths  of  a 
number  is  12f .     What  is  the  number  ? 

25.  If  T5^  of  the  goods  in  a  store  were  sold  for  $1000, 
what  was  the  value  of  the  whole  stock  of  goods  ? 

26.  If  3%  of  a  farm  is  worth  $1200,  what  is  the  value 
of  the  whole  farm  ? 


124  COMMON    FRACTIONS. 

Least  Common  Denominator. 

257.  Similar  Fractions.  Fractions  that  have  a  common 
denominator  are  called  similar  fractions. 

258.  Least  Common  Denominator.     The  L.  C.  M.  of 

the  denominators  of  a  series  of  fractions  is  called  their 
least  common  denominator.  The  letters  L.  C.  D.  stand  for 
the  words  Least  Common  Denominator. 

259.  Example.     Change  §,  j,  $  to  similar  fractions. 

The  L.  C.  D.  of  the  fractions  is  12. 

If  both  terms  of  $  are  multiplied  by  4,  the  value  of  the  fraction 
will  not  be  altered,  but  the  form  will  be  changed  to  ^. 

If  both  terms  of  J  are  multiplied  by  3,  the  equivalent  fraction  will 
be  ft. 

And  if  both  terms  of  $  are  multiplied  by  2,  the  equivalent  fraction 
will  be  |£. 

The  multipliers,  4,  3,  and  2,  are  obtained  by  dividing  12,  the  L.  C.  D. 
of  the  fractions,  by  the  respective  denominators  of  the  given  fractions. 
Therefore, 

260.  To  Change  Fractions  to  Similar  Fractions, 
Divide  the  least  common  denominator  of  the  fractions  by 

the  denominator  of  the  first  fraction,  and  multiply  both  terms 
of  this  fraction  by  the  quotient.  Proceed  in  the  same  way 
with  each  of  the  other  given  fractions. 

Change  J,  $,  \%  to  similar  fractions. 

4  =  22  ;  8  =  2«  ;  12  =  2*  X  3. 

Hence,  the  L.  C.  D.  is  28  X  3,  or  24. 


and 

5.  : 

=  ih 

result 

may 

be  written  as 

follows : 

18 

21 

22 

24 


COMMON    FRACTIONS.  125 

By  this  operation  the  parts  represented  by  the  given  fractions  have 
been  subdivided  into  smaller  parts  all  of  one  size,  and  the  numerators 
of  the  resulting  fractions  show  the  number  of  these  smaller  parts  con- 
tained in  the  given  fractions.  Thus,  the  quantities  denoted  by  f ,  |, 
and  |£  are  each  subdivided  into  24ths  of  the  unit,  and  contain  respec- 
tively 18,  21,  and  22  of  these  subdivisions. 

261.  Fractions  may  be  compared  by  first  reducing  them  to 
equivalent  fractions  having  a  common  denominator. 

Determine  the  greater  of  the  fractions  f  and  |£. 
In  this  case  the  least  common  denominator  is  112. 

Hence,  f  =  T8T\,  and   ft  =  f&. 

Therefore,  f  is  greater  than  |£. 

Exercise  57. 
Change  to  similar  fractions  : 

1.    h  h  f  10-    b  H>  H»  tt- 

11-    f>  f*  tVtI- 

13-  h  h  A>  A>  vfa- 

14.    h  A,  *>«>*■»• 

is-  i>  h  &Wi h  fr- 
it, ih  a,  H,i,  «, ». 

1^'     £'  |>  I'  A>  A>  A- 

9.  *>A>H>«-  i8-  A>  A>  ti»  H>  i>  H- 

19.  Which  is  the  greater,  ££  or  £|  ?  f  or  |  ?  §  or  $  ? 

20.  Arrange  the  fractions  T\,  |£,  £§  in  order  of  magni- 
tude. 

21.  Arrange  the  fractions  ^,  ^,  T\,  T7^  in  order  of 
magnitude. 

22.  Arrange  the  fractions  f,  J,  ^,  £§  in  order  of  mag- 
nitude. 


2. 

f >  f>  i>  A* 

3. 

fj  t>  A»  it* 

4. 

A»  A*  A>  A* 

5. 

H. «.  H.  II- 

6. 

|j  A»  A>  A>  ¥¥' 

7. 

tt>  A»  H>  1*.  H- 

8. 

h  h  H>  «• 

126  COMMON   FRACTIONS. 


Addition  of  Fractions. 
262.   Examples.     1.   Find  the  sum  of  fa  fa  fa. 
ixi+j  _5+l  +  7  _  ,,-j 


2.    Find  the  sum  of  J,  |,  £,  ^. 


Denominators  . 


4  =  22, 
6  =  2X3, 
9  =  32, 
12  =  22  X  3. 


Hence,  the  L.  C.  D.  =  22  x  32,  or  36. 

t  +  f  +  t+A****1**^**! 

=  W* 

=  2$. 

3.   Find  the  sum  of  4^,  2  fa  5^. 

f20=22X  5, 

Denominators  .  .  .W  15  =  3  x  5, 

[12  =  22X3. 

Hence,  the  L.  C.  D.  =  22  X  3  X  5  =  60. 

4H  +  2T^  +  5t\  =  ll*a±M±ai, 

=  12», 

=  13|§»     Hence, 

263.   To  Add  Fractions, 

Change  the  fractions  to  similar  fractions,  if  they  are  not 
similar,  and  write  the  sum  of  their  numerators  over  their 
common  denominator. 

If  any  of  the  expressions  to  be  added  are  whole  or  mixed 
numbers,  add  together  separately  the  fractions  and  the  in- 
tegers, and  find  the  sum  of  the  results. 


COMMON   FRACTIONS.  127 

Exercise  58. 
Find  the  sum  of  : 

1.  i  +  |.  6.  3i  +  j.  11.  |  +  f 

2.  £  +  §  +  £.         7.  2f  +  3|.  12.1  +  1. 

3.  i+i+|,         8.  l|  +  f.  13.  i+J. 

4.  li  +  2£.  9.  T*T  +  T3T  +  i  f  +  H-  14.  A  +  «■ 

5.  li+2|.  10.  8JV+6A  +  5«  +  H.       15.  A+H- 

16.  12f+7A-  IS-  i  +  i  +  i  +  i- 

17.  85A  +  27«.  19.  f+J+-|+|. 

20.  f  +  tt+A  +  A  +  M- 

21.  5H  +  11H  +  24H  +  A  +  17A  +  14  +  11A. 

22.  9t  +  15ii  +  163H  +  lH+10i. 

23.  3f  +  4f  +  lf +  2.  26.  4f  +  3f  +  2f  +  1J  +  A- 

24.  1A  +  2A  +  5A  +  A-     27.  H+^  +  10  +  fJ. 

25.  f  +  l|  +  2+3f+4TV     28.  li  +  ff+B+Afc  +  AV 

29.  2  +  §  +  l}  +  4f  +  5H. 

30.  3f +  6+.A  +  2A  +  5A  +  A- 

31-  A  +  A  +  3H  +  1H  +  2TWF- 

32.  A  +  A  +  9f  34.  Jl  +  H  +  H. 

33.  20A  +  HA  +  5i  +  305.  35.  A+ii  +  M  +  H- 

36.  317f  +  17/T  +  4A  +  A  +  6§  +  TV 

37.  4A  +  8A  +  4A+5?  +  6* +'§•■ 

38.  3a  +  5A  +  8^  +  f*  +  Mjfo. 

39.  4A  +  7A  +  6«  +  275A^  +  2«. 

40.  H  +  7A+6A  +  400A  +  51»f 

41.  3§  +  lJ  +  2J  +  3|  +  107A  +  2A. 

42.  5A  +  5f  +  2|  +  7A  +  12A. 

43.  4i  +  2j  +  3|  +  7J  +  8H- 

44.  6^  +  7§  +  8|+9|  +  8H- 

45.  7j  +  8}  +  5M  +  7H  +  9i- 

46.  5i  +  6|  +  7H  +  9H  +  3«  +  2i. 

47.  9}  +  10f  +  ll§  +  5jJ+7A  +  18f 

48.  H  +  A  +  if +  H  +  *  +  A  +  H. 


128  COMMON   FRACTIONS. 

Subtraction  of  Fractions. 

264.  Example.     From  \\  take  ft. 

The  L.  C.  D.  of  the  fractions  is  72. 

U-T5ff=iJTlP=H-     Hence, 

265.  To  Subtract  One  Fraction  from  Another, 

Change  the  fractions  to  similar  fractions,  if  they  are  not 
similar. 

Subtract  the  numerator  of  the  subtrahend  from  the  numer- 
ator of  the  minuend,  and  write  the  result  over  their  common 
denominator. 

266.  Examples.     1.    Subtract  3^  from  4|£. 
The  L.  C.  D.  of  the  fractions  is  120. 

If  the  terms  are  mixed  numbers,  subtract  separately  the  integers 
and  the  fractions,  and  unite  the  results. 

2.  Subtract  2£  from  5ft. 

6 A  "  2J  =  31^  =  2H1*1  =  2H- 

The  difference  between  5^  and  2  J  is  S^^1.  Since  we  cannot 
subtract  §  \  from  ££,  we  take  1,  that  is,  f  J,  from  the  3,  and  add  it  to 
£$,  making  f  J. 

3.  From  8  take  2§±. 

8  m  7ff,  and  7}|  -  2ft  =  6t^i  =  6 &. 

4.  Subtract  5J  from  15f . 

5J  +  £  =  6,  and  16|  +  |  =  15J. 
Then  15J  -  6  =  9J. 

Since  adding  the  same  number  to  both  the  minuend  and  the  sub- 
trahend does  not  alter  the  difference  between  them,  we  may  add 
to  the  subtrahend  such  a  fraction  as  will  make  it  a  whole  number,  pro- 
vided we  add  the  same  fraction  to  the  minuend. 


COMMON   FRACTIONS.  129 


Exercise  59. 

Find  the  value  of  : 

1.  52^-46. 

15.  7f-4f. 

29. 

*-!»«• 

2.  f-f. 

16.  6§  —  2j. 

30. 

1473-279H- 

3.  !-§• 

17.  9f-4f 

31. 

1473ft -279H 

4-  ft -ft- 

18.  4}-i. 

32. 

1473ft -279H- 

5.  «-ft. 

19.  6]  —  4f. 

33. 

278H  -30ft. 

6.  4  — £. 

20.  7£-2f. 

34. 

125ft -10iJ. 

7.  7-f. 

21.  8|-4f. 

35. 

118ft  -17ft. 

8.  3-f 

22.  85ft  — 27^. 

36. 

94ft -91«. 

9.  8-f 

23.  8ft -2«. 

37. 

7ft  -2«. 

10.  5-f. 

24.  10  — 3|. 

38. 

»*-«. 

11.  5-}. 

25.  120JJ  — 110H- 

39. 

H-tW 

12.  6J-5f 

26.  5«-|J. 

40. 

ft        3&V 

13.  4f  —  3f. 

27.  13ft -2jf 

41. 

ttf-lff- 

14.  74 -2ft.      28.  2HJ-1HJ.         42.  §«-«»• 

Plus  and  Minus  Terms. 

267.  Example.      Simplify  5|  -  4}  +  3§  -  2ft. 

5f  +  8f  =8i5rV-^  =  8!f  =  9Tv 

4|  +  2ft  =  6lftV±  =  6§f  =  7  ft, 

and  Oft  -  7ft  =  2**f#L  =  2ft. 

We  first  add  the  two  plus  terms  and  obtain  Oft. 
Then  we  add  the  two  minus  terms  and  obtain  7  ft. 
Then  we  subtract  the  sum  of  the  minus  terms  from  the  sum  of  the 
plus  terms  and  obtain  2ft.     Hence, 

268.  To  Simplify  an  Expression  of  Plus  and  Minus 
Terms, 

Subtract  the  sum  of  the  minus  terms  from  the  sum  of  the 
plus  terms. 


130  common  fractions. 

Exercise  60. 
Simplify : 

1.   3|-2|+4^  +  lJ-5A- 

2-    lA-tt  +  7*-2i-lH- 

3.  12-3»-l^-4/„  +  2iS-4S. 

4.  43A-li-M*-lH-2*i-2A-2H-8TV 

6-    (8A  +  U»  +  17«+«>)-(30j§  +  ll«). 

7.  (m«  +  93AV)  +  (172«  -  93^)- 

8.  (172H  +  93AV)-  ("2«  -93^V). 

»•  (ft-A)  +  (A +  !*«)• 
io    i-A-2S+»§  +  7A-i*-A- 

12.  9|  —  7  —  f  —  f. 

13.  5§  +  8f  —  If-4J. 

14.  6|-5§4-4§-4^. 

15.  HA  +  9J-6i-12*-3». 

16.  20§-2f-9|  +  10ft-141V 

17.  95§-9A-8»-14f  +  74j. 

18.  12}  +  23|-(4A  +  12f  +  7H). 

19.  16A  +  18A-(5|  +  9A  +  14A). 

20.  97f-(20  +  9|  +  18ft  +  24fJ). 

21.  2«  +  ajf-(l«  +  lJi  +  H). 

22.  «8  +  «J*-*«Wfc- 

Complex  Fractions. 

269.   Complex  Fractions.     A  fraction  that  has  a  frac- 
tion in  one  or  both  of  its  terms  is  called  a  complex  fraction. 

Thus,  i    i    ?i    2±j*f    <L±JJ  are  complex  fractions. 
7'   f    4  '   1  -  | '  4  X  A 

The  simplest  meaning  to  give  to  a  complex  fraction  is 
that  of  an  indicated  division.     Hence, 


3. 


COMMON   FRACTIONS.  131 

270.  To  Simplify  a  Complex  Fraction, 

Multiply  its  numerator  by  the  reciprocal  of  its  denominator. 

271.  Examples.     1.   |  =  f  -20  =  f  x±  =  §>. 

4f       6*7        6      U      12' 
2 

„2_    5_    6-5_    1.      5_    1  _  20  -  3  __  17 , 
3      h      l    9  V     *6      h  24  24' 

X9      24       ft       17       51        51 
3 

4j  — 3J-2J  +  1A      86  — 63-42  +  25=  6  =1 
3j  — 2f  +  2i  — A        64-48  +  45-7       54      9* 

A  complex  fraction  may  be  simplified  by  multiplying  both  its  terms 
by  the  smallest  number  that  will  make  them  integral.  This  multiplier 
will  always  be  the  L.  C.  D.  of  the  fractions  contained  in  the  terms  of 
the  given  fraction. 

Here  each  term  of  the  numerator  and  denominator  is  made  integral 
by  multiplying  it  by  18,  the  L.  C.  D.  of  the  fractions  contained  in  the 
numerator  and  denominator  of  the  given  fraction. 

6}-lt      63-lt  =  243  — 68     175      - 
*   ftoflf  H  35  35        v 

Here  the  compound  term  T\  of  If  is  first  reduced  to  the  simple 
term  §§,  and  then  the  numerator  and  denominator  of  the  resulting 
fraction  are  multiplied  by  36,  the  L.  C.  D.  of  the  small  fractions. 

Compound  terms  must  first  be  reduced  to  simple  terms. 


132  COMMON   FRACTIONS. 

Exercise  61. 
Simplify  : 

,24  4    ±  7    2J^zl3  1Q    6|-lft 

SJ-  8J-  •  l|-lf  •    2}+lf 

2    A  5    I*  «    10^~^  11      5*  +  2* 

7J-  •  8  A"  '  7} -3ft'  '  4§-3H 

17}  lit  of  3}    fl    ?Qf2A  12    §}_i 

*  13J-  "■  4J  of  ft-      ■  l|-i-2f"  •  14     If 

„     3iof5i  20    HXlf  +  iof2t-HX2 

,S-  Ui0I2j-  Hof2  +  Jof2J-lJoflJ 

•SI-2H'  1X6ft+7SXl*X9ft- 

16    2|  +  2j  22     8t-7f  +  5|-4) 

"  H-H'  »ft-8H+7l-«» 

"■StS-  M.JxJLxax^LxAxi|. 

„    M  +  lt  +  A  +  t  24    27_x87i      J 89ft 
H-H  +  A-f  374X98iX2iX  128- 

18  41-2}  .  26    iAx170^12j 
18'  6J  -2f  6ft*  399  *    7{  * 

19  m  -4^  +  31  /i_126  +  21V  31 
19     5f-4l  +  r           26-  (/     697  +  8lJ-5i- 


27. 
28. 
29. 
80. 


}ofl}|  +  l}of61-llof5l 

Jof2|of5§. 
HXtfrX  m  x  w 
5} 
ft  X  9ft  X  31  X  9ft 
ft  X  3ft  X  12}  X  m  X  ft 

21  X  7ft 
}  X  I  X  18f 


COMMON    FRACTIONS.  133 

To  Express  One  Number  as  the  Fraction  of  Another. 

272.  Examples.     1 .   What  fraction  of  8  is  7  ? 

Since  1  is  £  of  8, 

7  is  7  times  \  of  8  ; 
that  is,  7  is  |  of  8. 
Here  the  number  denoting  the  part  is  the  numerator  and  the  number 
denoting  the  whole  is  the  denominator  of  the  required  fraction. 

2.    What  fraction  of  |  is  f  ? 

Taking  the  number  denoting  the  part  for  the  numerator,  and  the 
number  denoting  the  whole  for  the  denominator,  we  have 

3  3       2       2 

| ;  and  this  becomes  -  X  -  =  -.     Therefore, 

§  16       1 

273.  To  Find  the  Fraction  that  One  Number  is  of 
Another, 

Take  the  number  denoting  the  part  for  the  numerator,  and 
the  number  denoting  the  whole  for  the  denominator. 

Exercise  62. 
What  fraction  of  : 

1.  8  is  3  ?  li.  2J  is  7|  ?  21.  $10  is  $§  ? 

2.  3  is  8  ?  12.  7^  is  2\  ?  22.  $100  is  $6  ? 

3.  9  is  7  ?  13.  3£  is  8}  ?  23.  $100  is  $4£  ? 

4.  7  is  9  ?  14.  $2  is  $l£?  24.  $4  is  $25  ? 

5.  8  is  12  ?  15.  $2£  is  $5  ?  25.  lOOf  is  8$  ? 

6.  12  is  8  ?  16.  $|  is  $J  ?  26.  21  is  {f  of  3|  ? 

7.  2£  is  |  ?  1 7.  $|  is  $|  ?  27.  18JJJ  is  f  of  33|  ?  . 

8.  £  is  2£  ?  18.  $2|  is  $*  ?  28.  3£  is  §  X  l£  ? 

9.  2j  is  11  ?  19.  $£  is  $TV?  29.  3TlT  X  5^7  is  1720  ? 
10.  li  is  2|  ?  20.  $1  is  $|  ?  30.  3£  X  f  of  $  is  If  ? 

What  part  of  : 

31-  U  X  £!is£  X4  x  V 

32.  13f  X  §  X  e\  is  }  of  1||  of  1 J  ? 

33.  H  +  «  +  A  +  t  i»  H  -  H  +  tV  ~  *  ? 


134 


COMMON    FRACTIONS. 


34.  4  *  -  2J  is  6£  -  2\  ? 

35.  17|-12fi85-A-A-*? 

36.  24-17Ai»7  +  A-A-a? 

37.  ^  of  2^  is  1§  -f-  2$  ? 

(«-F*Mi4-) 


13     ? 


Conversion  of  Fractions. 

274.  A  decimal  may  be  changed  to  a  common  fraction. 
Examples.     1.    Reduce  0.527  to  a  common  fraction. 

0.527  means  A  +  T^  +  T^  =  ^m±i  =  iWTT- 

2.  0.525  =  ^^  =  1*. 

3.  18.375  =  18  AWr  =  18j#  =  181.     Hence, 

275.  To  Reduce  a  Decimal  to  a  Common  Fraction, 
For  the  numerator,  write  the  figures  of  the  decimal ;  for 

the  denominator,  write  1  with  as  many  zeros  after  it  as  there 
are  figures  in  the  given  decimal. 


Exercise  63. 

Reduce  to  a  common  fraction  or  to  a 

mixed  number  : 

1. 

0.125. 

11. 

10.012575. 

21. 

0.603125, 

2. 

0.625. 

12. 

104.235. 

22. 

6.03125. 

3. 

0.675. 

13. 

50.0004. 

23. 

60.3125. 

4. 

10.864. 

14. 

100.001. 

24. 

7.0315. 

5. 

50.84. 

15. 

8.00725. 

25. 

12.0625. 

6. 

3.00025. 

16. 

20.018375. 

26. 

4.7168. 

7. 

8.1075. 

17. 

125.6048. 

27. 

0.0425. 

8. 

35.01024. 

18. 

0.128. 

28. 

6.46875. 

9. 

7.015625. 

19. 

0.73125 

29. 

0.00256. 

10. 

20.100256. 

20. 

1.1875. 

30. 

0.000375. 

COMMON   FRACTIONS.  135 

276.  A  common  fraction  may  be  reduced  to  a  decimal. 

Example.     Reduce  f  to  a  decimal. 

8)5.000 
0.625 

We  annex  zeros  to  the  numerator  of  the  fraction,  inserting  a  deci- 
mal point  before  the  zeros ;  and  then  divide  the  numerator  by  the 
denominator. 

By  this  operation  the  form  of  the  quotient  is  changed  from  f  to 
0.625,  but  the  value  remains  unchanged.     Hence, 

277.  To  Reduce  a  Common  Fraction  to  a  Decimal, 

Divide  the  numerator  by  the  denominator. 


Exercise  64. 

Reduce  to  a  decimal 

: 

i.  j. 

6.  t&* 

M.  Mv 

16. 

w- 

2-  «• 

'•    ^^ninS- 

12.  iM- 

17. 

foflf 

3-  A- 

8-  Mffo,. 

13-  *¥V- 

18. 

foffofA 

4-  *• 

»•  1MJ*- 

14-  AV- 

19. 

3f  of  4*. 

6.  «V 

10-  xfc- 

IS-  rtr 

Exercise  65. 

20. 

II  of  |f. 

Simplify  by  common  fractions,  then  by  reducing  the 
common  fractions  to  decimals,  and  show  that  the  results 
in  each  example  agree  : 

1  •  n  +  4f  +  9«  +  llff  8.  f |  -  4|. 

2.  84JJ  +  19H  +  H.  9.  82 1  -  37«. 

3.  m  +  13«  +  42fJ  +  2«  +  H-  10.  100  -  17J# 

4.  5 J  +  13*  +  19  A  +  7  A-  1 1  •  5*  ~  1*  of  1&. 

5.  5rfr  +  »ofl$  +  iof2*  +  |off  12.  H-H- 

6.  lAof2|.  13.  8J-Hof  A- 

7.  3^  +  2«.  14.   if  X  1000, 


136  COMMON    FRACTIONS. 

Repeating  Decimals. 

278.  If  the  denominator  of  a  common  fraction  in  its 
lowest  terms  contains  other  factors  than  2  and  5  (the  prime 
factors  of  10),  the  fraction  can  be  expressed  exactly  by  a 
decimal ;  otherwise  it  cannot. 

Thus,  if  we  take  the  fraction  ^r  to  express  as  a  decimal,  we  have 

0.27272727 and  the  division  will  never  end,  however  far  it  is 

carried. 

279.  A  decimal  that  contains  a  constantly  recurring 
figure  or  series  of  figures  is  called  a  repeating  decimal  or 
a  circulating  decimal. 

Thus,  the  decimal  0.27272727 is  a  repeating  decimal,  the  series 

of  figures  constantly  recurring  being  27. 

280.  Repetend.  The  figure  or  series  of  figures  that 
constantly  recurs  is  called  the  repetend. 

281.  If  the  repetend  begins  at  the  first  place  in  the 
decimal,  the  decimal  is  called  a  pure  repeating  decimal. 
If  the  repetend  begins  at  any  place  except  the  first,  it  is 
called  a  mixed  repeating  decimal. 

282.  In  writing  a  repeating  decimal,  we  place  dots 
over  the  first  and  last  figures  of  the  repetend. 

Thus,  we  write  0.272727 0.27,  and  we  write  0.333 0.3. 

283.  Examples.     1.   Reduce  £  to  a  decimal. 

The  denominator  contains  neither  2   nor  5  ;   the  first  figure  of 
the  decimal  begins  the  repetend  ;  and  we  reach  the 
0.857142    end  of  the  repetend  when  this  figure  appears  as  a 
7)6.000000    remainder.  0.857142  Ans. 


COMMON   FRACTIONS.  137 

2.    Reduce  12^f  to  a  decimal. 

0.53571428 

28)  15.0  We  reach  the  end  of  the  repetend  when  a 

140  previous  remainder  reappears. 

100  When  2  or  5  is  a  factor  of  the  denominator, 

84  the  number  of  decimal  places  preceding  the  repe- 

160 (1)    tend  is  equal  to  the   highest  exponent  which 

140  either  of  these  factors  has.     Thus,  28  is  equal 
200  to  22  X  7,  and  the  quotient  contains  two  deci- 
le mal  places  preceding  the  repetend. 
40  The  remainder  at  the  beginning  of  the  repe- 
28  tend  is  the  same  as  the  remainder  at  the  end 
120  of  the  repetend,   and  when    this    remainder 
112  reappears    we    need    carry    the    division    no 
80  further. 

56  Hence,  12|f  =  12.53571428. 
240 
224 
16  the  same  remainder  as  (1). 

We  may  shorten  the  work  by  cancelling  the  factor  4  common  to 
28  and  the  second  dividend  100,  giving  25  divided  by  7. 

Exercise  66. 
Reduce  to  a  decimal  : 

If  5.    3Jf  9.    9TW  13.    if. 

2.  T«T.  6.    2 A-  10.    llsY  14.    f7. 

3.  3TV  7.    ^.  11.    it.  15.    2jfr. 

4.  JJ.  8.    HiJ.  12.    ft.  16.    5A- 

rM     117       . 

17.  If    w7      03   1S   expressed   as  a  decimal,  how  many 

decimal  places  will  the  quotient  contain  ? 
119 

18.  If   o*  y  is   *s   exPresse(i  as  a  decimal,  how  many 

decimal  places  will  precede  the  repetend  ? 

57 

19.  If    -2      _    is    reduced    to    a   decimal,   how   many 

decimal  places  will  precede  the  repetend  ? 


138 


COMMON    FRACTIONS. 


To  Reduce  a  Repeating  Decimal  to  a  Common  Fraction. 

284.  Examples.     1.  Reduce  0.27  to  a  common  fraction. 

From  100  X  0.27,  or  27.2727 

take        1  X  0.27,  or  0.2727 

Then    99  X  0.27  =  27. 
Therefore,  0.27  =  |J  =  T3r. 

2.    Reduce  0.5243  to  a  common  fraction. 

From  10,000  X  0.5243,  or  5243.243243 

take       _10  X  0.5243,  or       5.243243 

Then     9990  X  0.5243  =  5243  —  6. 
Therefore,  0.5243  =  ^^  =  ||||  =  ^. 

We  multiply  the  decimal  by  such  a  number  as  will  put  the 
decimal  point  at  the  end  of  the  repetend,  then  by  such  a  number  as 
will  put  the  decimal  point  at  the  beginning  of  the  repeiend;  and  divide 
the  difference  of  the  products  by  the  difference  of  the  multipliers. 
Hence, 

285.  To  Reduce  a  Repeating  Decimal  to  a  Fraction, 
For  the  numerator,  write  the  difference  between  two  num- 
bers, one  expressed  by  the  figures  of  the  repetend,  and  the 
other  by  the  figures  that  precede  the  repetend. 

For  the  denominator,  write  a  9  for  each  figure  of  the  repe- 
tend, and  annex  a  0  for  each  figure  that  precedes  the  repe- 
tend. 

Exercise  67. 

Reduce  to  a  common  fraction  or  to  a  mixed  number : 


1. 

0.245. 

9. 

1.416. 

17. 

0.2368. 

2. 

0.425. 

10. 

0.5575. 

18. 

1.136. 

3. 

53.00243. 

11. 

2.08i. 

19. 

1.53i. 

4. 

7.20li. 

12. 

5.12297. 

20. 

3.28963. 

5. 

2.5306. 

13. 

0.3590. 

21. 

5.8783. 

6. 

0.00426. 

14. 

4.3i62. 

22. 

1.69408. 

7. 

31.203. 

15. 

0.7283. 

23. 

0.48324. 

8. 

0.351. 

16. 

5.142857. 

24. 

0.0012213. 

COMMON   FRACTIONS.  139 


The  Greatest  Common  Measure  and  the  Least  Common 
Multiple  of  Fractions. 

286.  If  we  divide  }  by  fa,  we  obtain  the  quotient  12. 
If  we  divide  f  by  },  we  obtain  the  quotient  5f . 

If  we  divide  f  by  ^85,  we  obtain  the  quotient  l£. 

We  see  from  these  three  examples  that  in  dividing  one 
fraction  by  another  the  quotient  is  integral  only  when  the 
numerator  of  the  divisor  is  a  measure  of  the  numerator  of 
the  dividend,  and  the  denominator  of  the  divisor  is  a  mul- 
tiple of  the  denominator  of  the  dividend. 

Therefore,  that  a  fraction  may  be  a  common  measure  of 
a  series  of  fractions,  its  numerator  must  be  a  measure  of 
each  numerator,  and  its  denominator  a  multiple  of  each 
denominator ;  and  that  a  fraction  may  be  the  greatest 
common  measure  of  a  series  of  fractions,  its  numerator 
must  be  the  greatest  common  measure  of  the  numerators 
of  the  fractions,  and  its  denominator  the  least  common 
multiple  of  the  denominators  of  the  fractions.     Hence, 

287.  To  Find  the  G.  C.  M.  of  a  Series  of  Fractions, 

Find  the  G.  C.  M.  of  the  numerators  for  the  numerator  of 
the  required  measure,  and  the  L.  C.  M.  of  the  denominators 
for  the  denominator  of  the  required  measure. 

288.  Conversely,  that  a  fraction  may  be  a  common  mul- 
tiple of  a  series  of  fractions,  its  numerator  must  be  a 
multiple  of  each  numerator,  and  its  denominator  a  measure 
of  each  denominator;  and,  that  a  fraction  may  be  the 
least  common  multiple  of  a  series  of  fractions,  its  numerator 
must  be  the  least  common  measure  of  the  numerators  of 
the  fractions,  and  its  denominator  the  greatest  common 
measure  of  the  denominators  of  the  fractions.     Hence, 


140  COMMON    FRACTIONS. 

289.  To  Find  the  L.  C.  M.  of  a  Series  of  Fractions, 

Find  the  L.  C.  M.  of  the  numerators  for  the  numerator  of 
the  required  multiple,  and  the  G.  C.  M.  of  the  denominators 
for  the  denominator  of  the  required  multiple. 

290.  Examples.     1.    Find  the  G.  C.  M.  of  ^,  y,  }f 

The  G.  C.  M.  of  5,  25,  35  =  5 ; 
and  the  L.  C.  M.  of  36,  9,  99  =  396. 
Hence,  j$5  is  the  G.  C.  M.  required. 

2.    Find  the  L.  C.  M.  of  ^,  y,  g  j. 

The  L.  C.  M.  of  5,  25,  35  =  175; 
and  the  G.  C.  M.  of  36,  9,  99  =  9. 
Hence,  a  J*  is  the  L.  C.  M.  required. 

Exercise  68. 
Find  the  G.  C.  M.  and  L.  C.  M.  of : 

1.  J,  «,  Jf.  7.   50^,67^,441,841,707. 

2.  2|f2f,A.  »•  hhhhhfv 

3.  33?,  50f.  9.  1A,  1»,  4f,  }f 

4.  A,Jf  |f  10-  18f,57f 

5.  5f  7Jf  81,  4f,  91,  6A-  It.  134f,  128J,  115f 
6-    fJ,J,i,  £,  A,  tV  12-  %%* 

13.  A,  B,  and  C  start  together  to  walk  in  the  same 
direction  round  a  circular  island.  It  takes  A  2%  days,  B 
2$  days,  C  2$  days  to  walk  round  the  island.  They  walk 
until  they  all  meet  at  the  point  of  starting.  In  how  many 
days  will  they  be  together  at  the  point  of  starting  ? 

14.  If  the  step  of  a  man  is  2^  feet,  and  that  of  a  horse 
is  2|  feet,  find  the  smallest  number  of  feet  which  is  an 
exact  number  of  steps  for  a  man  and  for  a  horse. 

15.  Find  the  largest  number  that  is  contained  without 
remainder  in  2f  6T7¥,  11£,  and  19£. 


COMMON   FRACTIONS.  141 


Exercise  69.  —  Review. 

1.  Simplify  JJ(|,  fifjjj,  T|«fF>  t»f J. 

2.  Which  is  greater,  and  by  how  much,  {  or  J  J  ? 

3.  Find  the  sum  of  3f,  2T*r,  6J,  7ft,  lft, 

4.  Simplify  5J  -  37  +  2ft  ~  If- 

5.  Simplify  If  +  3|  -  2ft  +  4  A  -  3ft. 

7      95^  16 

7.  Simplify  7-^2|;   j-J   jgj    15  ^f;    ^5    7ft -4-95 

6  A  #of4l  17 

8.  Simplify  7J|  X  8;  43*§  X  6f  ;  6*-t-8J;  5TV  X  51 ; 
»0f«i  jjofftofjofjofj;  «0fHfj  JXfXftXt 
Xf. 

9.  By  what  must  £  be  multiplied  to  obtain  ^  ?  ^  to 
obtain  §  ?  £  to  obtain  £  ?  §  to  obtain  $  ?  f  to  obtain  |  ? 

10.  By  what  must  £  be  divided  to  obtain  £  ?  §  to  obtain 
J  ?  {  to  obtain  $  ?  f  to  obtain  |?  8  to  obtain  7Jf  ? 

11.  What  number  exceeds  5§  by  4}  ? 

12.  Erom  what  must  6f  be  subtracted  to  leave  ^  of  3£  ? 

13.  What  fraction  falls  short  of  ft  by  ft  ? 

14.  What  fraction  must  be  added  to  ft  to  make  £}  Y 

15.  Reduce  to  decimals:    £;   J;   J;   j;   |;  |;  |;  jj  ft; 

fti    Tff>    TffJ    A>     H>    II J    t|5    &>    |>    7'    |>    ft  J    ft- 

16.  Reduce  to  common  fractions  :  0.16;  0.016;  0.125; 
0.13 ;  0.725  ;  0.625  ;  0.00625  ;  0.8125  ;  0.03125 ;  0.08  ;  0.54  ; 
0.016;  0.5437;  0.027;  0.277;  0.68494;  1.345. 

a.      fl.    2.8  of  2.27 

17.  Simplify - — • 

*     J       1.136 

18.  Multiply  6.954  by  5.303,  and  express  the  result  as  a 
whole  number  and  common  fraction. 


142  COMMON   FRACTIONS. 

19.  Simplify  1 J  of  2%  -f  6$  -=-  2§  and  reduce  the  result 
to  a  decimal. 

20.  From  what  number  can  4££  be  taken  9  times  and 
leave  no  remainder  ? 

21.  Of  what  fraction  is  17£  the  7th  part  ? 

22.  Add  f ,  0.35,  J,  |,  0.112,  45.28. 

23.  Reduce  to  decimals :   JJ;  ftj  gV;  JJ;  Jf;  ^. 

24.  What  part  of  }$  is  T2\T  ? 

25.  Divide  0.0015  by  0.012,  and  express  the  result  as  a 
common  fraction  in  its  lowest  terms. 

26.  Reduce  to  decimals :   fc  ssibin  H>  I* 

27.  The  product  of  two  factors  is  j,  and  one  factor  is  1 J  ; 
find  the  other  factor. 

28.  The  dividend  is  }£,  the  quotient  6^ ;  find  the  divisor. 

29.  The  dividend  is  12}£,  quotient  3,  remainder  1T^ ; 
find  the  divisor. 

30.  Find  the  G.  C.  M.  and  the  L.  C.  M.  of  833,  1127, 
1421,  343. 

31.  Arrange  in  order  of  magnitude  -ft-,  ££,  £$,  JJ,  §§. 

32.  Find  the  L.  C.  M.  of  jf,  $ f ,  ffs- 

33.  Find  the  G.  C.  M.  of  f  f ,  ^-,  |J,  and  6£. 

34.  Reduce  to  common  fractions  7.20li;  6.954. 

37.  Simplify  ^i—Al±^L. 

38.  Simplify  (3-71-1.908)  X  7.03 

2-2 -^A 

/         0.24  +  0.53\ 
40.   Simplify  li  of  2|  +  6J^  21+^  +  ^3^1 


COMMON   FRACTIONS.  143 

41.  Simplify  0.9  of  f  of  $  of  15f . 

42.  What  part  of  §  is  }  ? 

43.  What  part  of  0.390625  is  0.05  ? 

44.  What  fraction  of  0.2045  is  0.09  ? 

45.  Reduce  to  decimals  :   ff ;  ^f ;  f§. 

46.  The  G.  C.  M.  of  three  numbers  is  15,  and  their 
L.  C.  M.  is  450.     What  are  the  numbers  ? 

47.  A  merchant,  after  selling  5£  yards  and  3^  yards  from 
a  remnant  of  calico,  found  that  he  had  7§  yards  left.  What 
was  the  entire  length  of  the  remnant  ? 

48.  If  3|  yards  of  cloth  are  required  for  a  coat,  how 
many  coats  can  be  made  from  56^  yards  of  cloth  ? 

49.  A  grocer  bought  a  hogshead  of  sugar  weighing  744 
pounds  at  4}  cents  per  pound,  and  sold  it  at  5^  cents  per 
pound.     How  much  did  he  gain  ? 

50.  A  man,  after  selling  §  of  his  field,  sold  f  of  the 
remainder  and  then  had  13£  acres  left.  How  many  acres 
did  he  own  at  first  ? 

51.  A  railroad  train  passed  over  T7^  of  its  route  in  3£ 
hours.  In  how  many  hours  would  it  pass  over  the  entire 
route  ?     In  how  many  hours  over  §  of  the  route  ?  |  ?  T9¥  ? 

52.  A  boy,  being  asked  to  find  the  value  of  8-^  +  2^  + 
3§  -f-4§,  gave  as  his  answer  20.     How  great  was  his  error? 

53.  The  meter  is  equal  to  3^  feet,  very  nearly.  Express 
in  centimeters  the  value  of  4T^  feet. 

54.  For  a  piano  cover  a  lady  bought  2f  yards  of  plush 
at  $3^  per  yard,  the  same  amount  of  lining  flannel  at  $  J 
per  yard,  1^  yards  of  satin  at  #1J  per  yard,  and  l£  yards 
of  fringe  at  $1^  per  yard.  If  the  making  cost  $5,  what 
was  the  cost  of  the  piano  cover  ? 

55.  A  mason  built  6£  yards  of  wall  on  Monday,  4| 
yards  on  Tuesday,  if  yards  on  Wednesday,  and  7f  yards 
on  Thursday.  If  he  is  paid  $0.80  per  yard,  how  much 
has  he  earned  in  the  four  days  together  ? 


144  COMMON   FRACTIONS. 

56.  A  coal  dealer  sold  100  tons  of  coal.  If  he  shipped 
by  six  cars  14£.  14^,  14-^,  14385,  14T7ff,  14^  tons  respec- 
tively, how  many  tons  must  he  load  on  the  seventh  car  to 
complete  his  shipment  ? 

57.  The  moon's  diameter  is  -ft  that  of  the  earth,  and 
the  sun's  diameter  is  110  times  that  of  the  earth.  What 
fraction  of  the  sun's  diameter  is  the  moon's  diameter? 

58.  If  a  silver  rupee  in  Calcutta  is  worth  $£§,  what  is 
the  value  in  dollars  and  cents  of  a  fan  costing  4$  rupees  ? 

59.  If  a  man  can  do  -ft-  of  a  piece  of  work  in  25  days, 
what  fraction  of  the  work  can  he  do  in  62f  days  ? 

60.  I  paid  a  tailor  $3£  a  yard  for  5£  yards  of  broad- 
cloth. On  measuring  it,  I  found  that  there  were  only  4| 
yards.     How  much  money  ought  the  tailor  to  return  ? 

61.  From  a  tank  full  of  water  §  of  the  water  was  drawn 
otf.  Then  35  gallons  were  added,  and  the  tank  was  just 
half  full.     What  is  the  capacity  of  the  tank  ? 

62.  What  number  exceeds  the  sum  of  its  fourth,  fifth, 
sixth,  and  seventh  parts  by  101  ? 

63.  A  trader  bought  wheat  at  75  cents  a  bushel,  and 
sold  it  at  71  cents  a  bushel.  How  many  cents  did  he  lose 
on  every  dollar  he  paid  ? 

64.  How  many  bushels  of  potatoes  at  $£  per  bushel  will 
pay  for  16  bushels  of  wheat  at  $£#  per  bushel  ? 

65.  From  a  piece  of  calico  containing  35$  yards,  there 
have  been  sold  at  different  times  12|  yards,  2\  yards,  2^\ 
yards,  and  8|  yards.     How  many  yards  remain  ? 

66.  If  gun  metal  is  composed  of  90£  parts  of  copper  to 
9^  parts  of  tin  by  weight,  how  many  ounces  of  tin  are  there 
in  one  pound  (16  ounces)  of  gun  metal  ?  how  many  ounces 
of  copper  in  one  pound  ? 

67.  One  man  mows  \  of  a  field,  a  second  ^  of  it,  and  a 
third  ^  of  it.  What  fraction  of  the  field  remains  to  be 
mowed  ? 


COMMON   FRACTIONS.  145 

68.  Bell  metal  by  weight  consists  of  4  parts  of  copper 
to  1  part  of  tin.  What  is  the  cost  of  a  bell  weighing  12,400 
pounds,  if  the  copper  costs  19  cents  per  pound,  the  tin  22^ 
cents  per  pound,  and  the  cost  of  making  is  $500  ?     , 

69.  If  an  ore  loses  i£  of  its  weight  in  roasting,  and  -^ 
of  the  remainder  in  smelting,  how  many  tons  of  ore  must 
be  mined  to  obtain  466  tons  of  pure  metal  ? 

70.  The  amount  of  starch  in  potatoes  is  J  J-  of  their 
weight,  but  the  amount  that  can  usually  be  extracted  is 
only  T^.  How  many  pounds  of  starch  can  be  obtained 
from  100  pounds  of  potatoes,  and  how  many  pounds  of 
starch  will  be  left  in  the  potatoes  ? 

71.  How  many  pairs  of  trousers,  each  pair  requiring  2| 
yards,  can  be  made  from  33£  yards  of  cloth  ? 

72.  If  3^  yards  of  cloth  are  required  for  a  shirt,  how 
many  shirts  can  be  made  from  12  pieces  of  cloth,  each 
piece  measuring  47£  yards  ? 

73.  Green  coffee  when  roasted  loses  £  of  its  weight.  If 
a  dealer  buys  green  coffee  at  22\  cents  a  pound,  and  sells 
it  roasted  at  30  cents  a  pound,  what  will  be  his  gain  in 
selling  1000  pounds  of  roasted  coffee,  the  cost  of  roasting 
the  whole  quantity  being  $ 2.25  ? 

74.  If  an  iron  bar,  when  heated  1  degree,  expands 
T^sfftf  of  its  length,  what  is  the  length  at  212  degrees  of 
a  bar  whose  length  at  32  degrees  is  10$  feet  ? 

75.  If  a  horse  eats  -^  of  a  ton  of  hay  in  30  days,  how 
long  will  4^  tons  of  hay  last  5  horses  ? 

76.  If  4  is  added  to  both  terms  of  the  fraction  |£,  by 
how  much  is  the  value  of  the  fraction  increased  ? 

77.  If  4  is  subtracted  from  both  terms  of  the  fraction 
\%,  by  how  much  is  the  value  of  the  fraction  decreased  ? 

78.  Find  the  least  number  of  apples  that  arranged  in 
groups  of  8,  9,  10,  or  12  will  have  just  6  over  in  each 
case. 


146  COMMON    FRACTIONS. 

79.  The  diameter  of  a  bicycle  wheel  is  2$  feet,  and  the 
circumference  is  3\  times  the  diameter.  How  many  times 
does  the  wheel  turn  in  going  1  mile  (5280  feet)? 

80.  What  is  the  least  number  of  yards  of  carpet  in  a 
roll  that  can  be  cut  into  lengths  of  exactly  13£  yards,  8 
yards,  or  11^  yards  ? 

81.  What  is  the  length  of  the  longest  pole  that  will 
exactly  measure  the  sides  of  a  field  whose  lengths  are 
respectively  135|  yards,  118$  yards,  152  yards,  and  202§ 
yards  ? 

82.  Find  the  least  multiplier  of  £,  tf,  and  f  f  that  will 
make  each  product  an  integral  number. 

83.  Find  the  least  integral  number  that  is  exactly  divisi- 
ble by  5±,  3£,  and  7. 

84.  Four  bells  commence  tolling  together,  and  toll  at 
intervals  of  1,  1|,  1^,  and  l-^-  seconds,  respectively.  In 
how  many  seconds  will  all  four  toll  again  at  the  same 
instant  ? 

85.  What  number  multiplied  by  fT  of  ft  of  29$  will 
give  102$  for  the  product  ? 

86.  How  many  miles  an  hour  must  a  man  walk  to  go 
28  miles  in  7T75  hours  ? 

87.  If  the  rent  of  5^  acres  of  land  is  $21§,  what  will 
be  the  rent  of  19^  acres  at  the  same  rate  ? 

31i 

88.  If  the  English  acre  is  -r-r  of  an  Irish  acre,  how 

4y 

many  English  acres  are  there  in  218f  Irish  acres  ? 

89.  Resolve  the  denominator  of  ff  into  its  prime  fac- 
tors ;  from  the  result  state  the  number  of  figures  the 
equivalent  decimal  will  have,  and  the  number  that  will 
precede  the  repetend. 

90.  Find  the  greatest  common  measure  of  9083,  9207, 
8897. 


CHAPTER  VIII. 
COMPOUND  QUANTITIES. 

291.  A  quantity  expressed  in  units  of  one  denomination 
is  called  a  simple  quantity.  A  quantity  expressed  in  units 
of  two  or  more  denominations  is  called  a  compound  quan- 
tity, or  a  compound  denominate  number. 

Thus,  4£  gallons  is  a  simple  quantity  ;  but  its  equivalent,  4  gallons 
1  quart,  is  a  compound  quantity. 

292.  Reduction.  The  process  of  changing  the  denomina- 
tion in  which  a  quantity  is  expressed,  without  changing 
the  value  of  the  quantity,  is  called  reduction. 

Pleasures  of  Capacity. 

293.  Liquid  Measure.  Liquid  measure  is  used  in 
measuring  liquids,  as  water,  milk,  etc. 

Table. 

4  gills  (gi.)  =  1  pint  (pt.). 
2  pints         =  1  quart  (qt.). 
4  quarts       =  1  gallon  (gal.). 
1  gal.  =  4  qt.  =  8  pt.  =  32  gi.     1  gal.  contains  231  cubic  inches. 


31£  gal.       =  1  barrel  (bbl.). 
63    gal.        =  1  hogshead  (hhd.). 

Note.  Casks  holding  from  28  gal.  to  43  gal.  are  called  barrels, 
and  casks  holding  from  54  gal.  to  63  gal.  are  called  hogsheads. 

Whenever  barrels  or  hogsheads  are  used  as  measures,  a  barrel 
means  31£  gallons,  and  a  hogshead  means  63  gallons. 


148  COMPOUND   QUANTITIES. 

294.  Dry  Measure.  Dry  measure  is  used  in  measuring 
dry  articles,  as  grain,  seeds,  fruit,  vegetables. 

Table. 

2  pints  (pt.)  =  1  quart  (qt.). 
8  quarts  =  1  peck  (pk.). 
4  pecks         =  1  bushel  (bu.). 

1  bu.  =  4  pk.  =  32  qt.  1  bu.  contains  2150.42  cubic  inches. 

Note.  In  measuring  grain,  seeds,  and  small  fruits,  the  measure 
must  be  even  full.  In  measuring  apples,  potatoes,  and  other  large 
articles,  the  measure  must  be  heaping  full. 

The  quart  of  liquid  measure  contains  67 f  cubic  inches,  and  the 
quart  of  dry  measure  671  cubic  inches. 

In  Great  Britain,  the  quart  of  liquid  measure  is  of  the  same  size  as 
the  quart  of  dry  measure,  and  contains  69.3185  cubic  inches.  There- 
fore, the  imperial  gallon  of  Great  Britain  contains  277.274  cubic 
inches,  and  the  imperial  bushel  contains  2218.192  cubic  inches. 

The  quarter  contains  8  imperial  bushels. 


Reduction  of  Compound  Quantities. 

295.  If  the  reduction  is  from  a  higher  denomination  to 
a  lower,  it  is  called  reduction  descending.  If  the  reduction 
is  from  a  lower  denomination  to  a  higher,  it  is  called  reduc- 
tion ascending. 

Thus,  2  pecks  =  16  quarts  is  an  example  of  reduction  descending  ; 
and  24  quarts  =  6  gallons  is  an  example  of  reduction  ascending. 

296.  Example.     Reduce  12  gal.  2  qt.  1  pt.  to  pints. 

gal.  qt.       pt.       Solution.     12  gal.  =  12  X  4  qt.  =  48  qt,  and  48 

12  2         1     qt.  with  the  2  qt.  added  are  50  qt. 

_4  50  qt.  =  50  X  2  pt.  =  100  pt.,  and  100  pt.  with 

60  the  1  pt.  added  are  101  pt. 

2  The  multiplicand  and  multiplier  are  interchanged 

101  in  the  operation,     Hence, 


COMPOUND   QUANTITIES.  149 

297.  In  Reduction  Descending, 

Multiply  the  given  number  of  units  of  the  highest  denomi- 
nation by  the  number  of  units  of  the  next  lower  denomination 
required  to  make  one  of  this  higher  ;  and  add  to  the  -product 
the  given  number  of  units  of  this  lower  denomination. 

Proceed  in  this  way  with  each  successive  result  until  the 
required  denomination  is  reached. 

298.  Example.     Reduce  221  pt.  to  higher  units. 

2  221  pt.  Solution.     221   pt.  =  110  qt.    and   1   pt. 

4  110  qt.  .  .  .  1  pt.     over.      110  qt.  =  27    gal.    and  2   qt.   over. 
27  gal.    .  .  2  qt.     Therefore,   221    pt.  =  27    gal.   2   qt.    1   pt. 
Hence, 

299.  In  Reduction  Ascending, 

Divide  by  the  number  of  units  required  to  make  one  of  the 
next  higher  denomination. 

Divide  this  quotient  and  each  successive  quotient  in  like 
manner  until  the  required  denomination  is  reached. 

The  last  quotient  with  the  several  remainders,  arranged  in 
order,  is  the  answer  sought. 

Exercise  70. 

Reduce :     ■ 

1.  3  pk.  5  qt.  1  pt.  to  pints. 

2.  4234  pt.  (dry  measure)  to  higher  units. 

3.  24  gal.  2  qt.  1  pt.  2  gi.  to  gills. 

4.  3047  gills  to  higher  units. 

5.  1715^  bu.  to  pints. 

6.  508  dry  quarts  to  higher  units. 

7.  1016  liquid  pints  to  higher  units. 

8.  44  gal.  3  qt.  1  pt.  to  pints. 

9.  44  bu.  3  pk.  7  qt.  1  pt.  to  pints. 

10.  272  liquid  quarts  to  dry  quarts. 

1 1 .  429  dry  quarts  to  liquid  quarts. 


Solution. 

V 

gal.        qt. 

pt 

3        2 

1 

5        3 

0 

13        1 

1 

9        0 

1 

150  COMPOUND   QUANTITIES. 


Addition  and  Subtraction  of  Compound  Quantities. 

300.   Examples.     1.    Add  3  gal.  2  qt.  1  pt. ;  5  gal.  3  qt. ; 
13  gal.  1  qt.  1  pt.;  9  gal.  1  pt. 

We  write  units  of  the  same  name  in  the  same  column, 
and  add  the  columns,  beginning  with  the  pints. 
3  pt.  =  1  qt.  and  1  pt.  over.  We  write  the  1  pt. 
under  the  pints,  and  add  the  1  qt.  to  the  quarts. 
7  qt.  =  1  gal.  and  3  qt.  over.  We  write  the  3  qt. 
under  the  quarts,  and  add  the  1  gal.  to  the  gallons. 
31        3        1    The  required  sum  is,  therefore,  31  gal.  3  qt.  1  pt. 

2.   From  5  bu.  2  pk.  3  qt.  take  3  bu.  3  pk.  3  qt. 

Solution.     3  qt.  —  3  qt.  =  0  qt.     We  write  0  under  the  quarts. 

Since  we  cannot  take  3  pk.  from  2  pk.,  we  take 
bu.  pk.  qt.  1  bu.  (4  pk.)  from  the  6  bu.  and  add  it  to  the  2  pk. 
5  2  3  4  pk.  +  2  pk.  =  6  pk.,  and  6  pk.  -  3  pk.  =  3  pk. 
3  3  3  We  write  3  under  the  pecks.  Then  4  bu.  —  3  bu. 
1        3        0      =1  bu.      The  required  difference  is,   therefore, 

1  bu.  3  pk. 


Exercise  71. 

1.  Add  5  bu.  3  pk.  6  qt.  1  pt.;  6  bu.  2  pk.  7  qt.;  7  bu. 
1  pk.  1  qt.  1  pt. ;  1  pk.  7  qt. ;  2  bu.  3  pk.  1  pt. 

2.  Add  50  gal.  3  qt.  1  pt.  3  gi.;  12  gal.  1  qt.  1  pt.  1  gi.; 
5  gal.  2  qt.  1  pt.  2  gi.;  75  gal.  3  qt.  1  pt.  3  gi.;  80  gal.  3 
qt.  1  gi. ;  17  gal.  1  qt.  1  pt.  3  gi. 

3.  Add  4  gal.  3  qt.  1  pt.;  3  gal.  2  qt.  1£  pt.;  12  gal.  3 
qt.;  14  gal.  l£j>t.;  5  gal.  2  qt.  1  pt. 

4.  Subtract  5  bu.  1  pk.  6  qt.  1  pt.  from  5  bu.  3  pk.  3  qt. 

5.  Subtract  22  gal.  3  qt.  1  pt.  from  30  gal.  2  qt. 

6.  Add  6  bu.  1  pk.  7  qt.  1  pt.;  2  bu.  2  pk.  5  qt.  -j-  pt.; 
19  bu.  3  pk.  0  qt.  1  pt.;  14  bu.  2  pk.  4  qt.  1£  pt.;  10  bu.  1 
pk.  3  qt.  1  pt.;  5  bu.  3  pk.  2  qt. 

7.  Find  the  difference  between  2  bu.  and  5  qt.  1  pt 


COMPOUND   QUANTITIES.  151 

Multiplication  and  Division  of  Compound  Quantities. 

301.  Examples.     1.    Multiply  15  gal.  3  qt.  1  pt.  by  5. 

Solution.     5  X  1  pt.  =  5  pt.  =  2  qt.  1  pt.  We  write  the  1  pt.  under 

the  pints,  and  reserve  the  2  qt.  to  be  added  to  5  X 

gal.       qt.      pt.   3  qt.     5X3  qt.  =  15  qt.,  and  this  with  the  2  qt. 

15        3         1     reserved  =  17  qt.  =  4  gal.  1  qt.     We  write  the  1  qt 

5    under  the  quarts,  and  add  the  4  gal.  to  5  X  15  gal., 

79        1        1     having  79  gal.     The  required  product  is,  therefore, 
79  gal.  1  qt.  1  pt. 

2.   Divide  122  bu.  2  pk.  7  qt.  1  pt.  by  5. 

Solution.     122  bu.  -f-  5  =  24  bu.  and   2  bu.  over.     We  write  24 

under  the  bushels.     2  bu.  =2x4  pk.  =  8 

bu.       pk.       qt.       pt.  pk. ,  and  8  pk.  +  2  pk.  =  10  pk.    10  pk.  -r  5 

5)122        2        7        1     =  2pk.     We  write  2  under  the  pecks.     7  qt. 

24        2        1         1     -=-  5  as  1  qt.   and  2  qt.  over.      We  write  1 

under  the  quarts.    2  qt.  =  2  X  2  pt.  =  4  pt., 

and  4  pt.  +  1  pt.  =  5  pt.     5  pt.  -J-  5  =  1  pt.      We  write*  1  under  the 

pints.     The  required  quotient  is,  therefore,  24  bu.  2  pk.  1  qt.  1  pt. 

Exercise  72. 

1.  Multiply  19  gal.  3  qt.  1  pt.  by  70. 

2.  Multiply  43  bu.  2  pk.  6  qt.  1  pt.  by  63. 

3.  Multiply  17  bu.  3  pk.  6  qt.  by  8. 

4.  Multiply  26  gal.  2  qt.  1  pt.  3  gi.  by  12. 

5.  Multiply  12  bu.  3  pk.  7  qt.  1  pt.  by  25. 

6.  Divide  34  gal.  3  qt.  1  gi.  by  7. 

7.  Divide  147  gal.  2  qt.  1  pt.  2  gi.  by  17. 

8.  Divide  54  bu.  3  pk.  2  qt.  1  pt.  by  11. 

9.  Divide  34  bu.  3  pk.  5  qt.  1  pt.  by  15. 

302.  All  compound  quantities  are  reduced,  added,  sub- 
tracted, multiplied,  divided,  by  the  methods  given  for 
measures  of  capacity. 


152  COMPOUND   QUANTITIES. 


Measures  of  Weight. 

303.  Troy  Weight.  Troy  weight  is  used  in  weighing 
gold,  silver,  and  precious  stones. 

Table. 

24  grains  (gr.)      =  1  pennyweight  (dwt.). 
20  pennyweights  —  1  ounce  (oz.). 
12  ounces  =  1  pound  (lb.). 

The  pound  troy  contains  5760  grains. 

304.  Avoirdupois  Weight.  Avoirdupois  weight  is  used 
in  weighing  all  articles  except  gold,  silver,  and  precious 
stones. 

Table. 

16  ounces  (oz.)  =  1  pound  (lb.). 
100  pounds         =  1  hundredweight  (cwt.). 
2000  pounds         =  1  ton  (t.). 


112  pounds  =  1  long  hundredweight. 
2240  pounds  =  1  long  ton. 

The  long  ton  is  used  in  the  United  States  Custom  Houses,  and  in 
wholesale  transactions  in  iron  and  coal. 

The  pound  avoirdupois  contains  7000  troy. grains. 

Exercise  73. 

1.  Reduce  27,587  gr.  to  higher  troy  units. 

2.  Reduce  34,652  pounds  avoirdupois  to  long  tons,  etc. 

3.  Reduce  136,851  ounces  avoirdupois  to  higher  units. 

4.  Reduce  864,205  gr.  to  higher  troy  units. 

5.  Reduce  864,205  gr.  to  higher  avoirdupois  units. 

6.  Reduce  5  lb.  7  oz.  6  dwt.  12  gr.  to  grains. 

7.  Reduce  745  lb.  avoirdupois  to  troy  measures. 

8.  Reduce  745  lb.  troy  to  avoirdupois  measures. 

9.  Reduce  1,440,445  oz.  avoirdupois  to  higher  units. 


COMPOUND   QUANTITIES.  153 

10.  Reduce  5,640,773  oz.  avoirdupois  to  higher  units. 

11.  Add  48  t.  13  cwt.  75  lb.  6  oz.;  25  t.  12  cwt.  27  lb. 
8  oz.;  51  t.  10  cwt.  44  lb.;  80  t.  5  cwt.  6  oz.;  19  cwt.  27 
lb.;  25  1b.  8  oz.;  5  t.  5  cwt.  5  1b. 

12.  Add  13  lb.  4  oz.  8  dwt.  6  gr.;  25  lb.  8  oz.  13  dwt. 
20  gr.;  8  lb.  11  oz.  14  gr.;  20  lb.  16  dwt.  8  gr.;  15  lb.  9 
oz.  12  dwt. ;  4  oz.  3  dwt. 

13.  Subtract  23  lb.  8  oz.  19  dwt.  10  gr.  from  58  lb.  6  oz. 
17  dwt.  21  gr. 

14.  Subtract  17  t.  7  cwt.  17  lb.  6  oz.  from  25  t.  13  cwt. 
15  lb.  12  oz. 

15.  Multiply  3  lb.  4  oz.  8  dwt.  10  gr.  by  10. 

16.  Multiply  5  t.  10  cwt.  67  lb.  4  oz.  by  15. 

17.  Divide  17  t.  19  cwt.  79  lb.  8  oz.  by  8. 

18.  Divide  60  lb.  6  oz.  10  dwt.  20  gr.  by  7. 

19.  How  many  bags  each  holding  2  bu.  1  pk.  3  qt.  are 
required  to  hold  234  bu.  1  pk.  4  qt.  of  corn  ? 

Note.    Reduce  both  quantities  to  quarts. 

20.  What  is  the  value  at  4 J  cents  a  pound  of  a  calf 
weighing  184  lb.  6  oz.? 

21.  How  many  tablespoons  each  weighing  2  oz.  17  dwt. 
12  gr.  can  be  made  from  155  oz.  5  dwt.  of  silver  ? 

305.  In  compounding  medicines,  apothecaries  make  use 
of  the  following  : 

Apothecaries'  Weight. 

20  grains  (gr.)  =  1  scruple  O). 

3  scruples       =  1  dram  (5). 

8  drams  =  1  ounce  (§  ). 

12  ounces         =  1  lb.  troy. 

Apothecaries'1  Measure. 
60  minims  (m)  =  1  dram  (m  lx.). 
8  drams  =  1  ounce  (fl.  drm.  viij.). 

16  ounces  =  1  pint  (fl.  oz.  xvj.). 


154  COMPOUND    QUANTITIES. 


Measures  of  Length. 

306.  The  unit  of  measure  for  lengths  is  the  yard.  From 
the  yard  are  derived  the  units  of  surface  and  volume. 

307.  The  standard  yard  of  Great  Britain,  as  defined  by 
Act  of  Parliament,  is  the  distance  between  the  centres  of 
two  cylindrical  holes  in  a  certain  bar  of  gun  metal,  when 
the  metal  has  a  temperature  of  62  degrees  Fahrenheit. 

The  standard  yard  of  the  United  States  conforms  as 
nearly  as  possible  to  that  of  Great  Britain. 

308.  Measures  of  length  are  used  in  measuring  lines 
or  distances. 

Table. 

12  inches  (in.)  =  1  foot  (ft.). 

3  feet  =  1  yard  (yd.). 

5£  yards,  or  l(ty  feet  =  1  rod  (rd.). 
320  rods  =  1  mile  (mi.). 

1  mi.  =  320  rd.  =  1760  yd.  =  6280  ft. 

Note.  A  hand  (used  in  measuring  the  height  of  horses)  =  4  in. ; 
a  knot  (used  in  navigation)  =  6086  ft. ;  a  league  =  3  knots  ;  a  fathom 
=  6  ft. ;  a  cable  length  =120  fathoms  ;  a  line  =  ^  in. ;  a  barleycorn 
=  \  in. ;  a  palm  =  3  in.;  a  span  s 9  in. ;  a  cubit  =  18  in. ;  a  military 
pace  =  28  in. ;  a  furlong  =  \  mi. 

309.  Examples.  1.  Change  106,760  ft.  to  higher  de- 
nominations. 

Solution.     There  are  16£  ft.,  or  33  half-feet,  in  a  rod ;  so  we 

change  the  106,760  ft,    to 
16^)106760  ft.  half-feet,  and  these  to  rods, 

2  by   dividing    by   33.      The 

33)213520  half-feet.  remainder  is   10  half-feet, 

320)6470  rd.    ...  10  half-feet  =  5  ft.     or  5  ft.     6470  rd.  =  20  mi. 

20  mi.   ...  70  rd.  70  rd.     Therefore,  106,760 

ft.  =  20  mi.  70  rd.  5  ft. 


COMPOUND   QUANTITIES.  155 

2.    Add  4  mi.  110  rd.  5  yd.  1  ft.  8  in.  and  6  mi.  25  rd. 
4  yd.  1  ft.  6  in. 


mi. 

rd. 

yd. 

ft.    in. 

4 

110 

5 

1       8 

6 

25 

4 

1     6 

Solution.     We  have  for  the  sum  10  mi. 

136  rd.  3^  yd.  0  ft.  2  in.     We  reduce  the  \ 

10     136     M    0    2        yd.  of  this  sum  and  add  its  value  1  ft.  6  in. 

16        to  the  0  ft.  2  in.     We  have  for  the  sum, 

10     136    3      1     8        therefore,  10  mi.  136  rd.  3  yd.  1  ft.  8  in. 


Exercise  74. 

1.  Keduce  3  yd.  2  ft.  to  inches. 

2.  Eeduce  4  mi.  124  rd.  3  yd.  2  ft.  to  feet. 

3.  Eeduce  27  rd.  4  yd.  9  in.  to  inches. 

4.  Eeduce  290  leagues  to  feet. 

5.  Eeduce  82,976,432  in.  to  higher  units. 

6.  Eeduce  7  mi.  3  yd.  1  ft.  6  in.  to  inches. 

7.  Eeduce  22  mi.  222  rd.  4  ft.  8  in.  to  inches. 

8.  Eeduce  712  mi.  to  feet. 

9.  Eeduce  540,451  ft.  to  higher  units. 

10.  Eeduce  271,256  in.  to  higher  units. 

11.  Eeduce  723,964  ft.  to  higher  units. 

12.  Eeduce  233,205  in.  to  higher  units. 

13.  How  many  feet  high  is  a  horse  16  hands  high  ? 

14.  Add  6  mi.  120  'rd.  3  yd.  2  ft.  2  in.;  18  mi.  15  rd. 
1  yd.  1  ft.  6  in.;  3  mi.  215  rd.  2  yd.  2  ft.  3  in.;  7  mi. 
95  rd.  1  yd.  1  ft.  8  in. 

15.  Subtract  3  mi.  217  rd.  4  yd.  1  ft.  3  in.  from  4  mi. 
100  rd.  3  yd.  2  in. 

16.  Multiply  5  mi.  126  rd.  9  ft.  6  in.  by  7125. 

17.  Divide  54  mi.  124  rd.  1  yd.  2  ft.  6  in.  by  33. 

18.  If  a  man  builds  1  rd.  1  yd.  1  ft.  6  in.  of  stone  wall 
in  one  day,  how  much  will  he  build  in  26  days  ? 

19.  A  man  builds  25  rd.  2  yd.  1  ft.  6  in.  of  wall  in  20 
days.     How  much  does  he  build  per  day  ? 


156  COMPOUND   QUANTITIES. 


Measures  of  Surface. 

310.   The  area  of  a  surface  is  the  number  of  square 
units  it  contains. 


Table 

144  square 


A  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.). 
9  square  feet  =  1  square  yard  (sq.  yd.). 

304;  square  yards,  or  )  ,  _  t        _  % 

272*  square  feet  \         =  1  «l^e  rod  (sq.  rd.). 

160  square  rods  or  ) 
43,560  square  feet        }  -  1  acre  (A.). 

640  acres  =  1  square  mile  (sq.  mi.). 

Note.     In  measures  of  surface  the  scale  ascends  and  descends  by 
the  squares  of  the  units  of  length  ;  thus,  144  =  122  ;  9  =  32  ;  30±  = 

(5£)2;272±=(16£)2. 

Exercise  75. 

1.  Reduce  92,638  sq.  yd.  to  square  inches. 

2.  Reduce  1,223,527  sq.  in.  to  higher  units. 

3.  Reduce  721  sq.  mi.  to  square  rods. 

4.  Reduce  34,729  sq.  yd.  to  higher  units. 

5.  Reduce  to  square  inches  3  A.  107  sq.  rd.  27  sq.  yd. 
7  sq.  ft.  23  sq.  in. 

6.  Reduce  99,894,712  sq.  in.  to  higher  units. 

7.  Reduce  15,376  sq.  yd.  to  higher  units. 

8.  Reduce  562,934  sq.  in.  to  higher  units. 

9.  Add  74  A.  21  sq.  rd.  5  sq.  yd.  4  sq.  ft.  100  sq.  in.; 
123  A.  23  sq.  rd.  13  sq.  yd.  5  sq.  ft.  83  sq.  in.;  112  A.  106 
sq.  rd.  17  sq.  yd.  8  sq.  ft.  7  sq.  in.;  541  A.  50  sq.  rd. 
23  sq.  yd.  24  sq.  in. 

10.  From  20  A.  take  13  A.  150  sq.  rd.  98  sq.  ft.  10  sq.  in. 

11.  Multiply  27  A.  76  sq.  rd.  22  sq.  yd.  5  sq.  ft.  by  90. 

12.  Divide  74,128  sq.  mi.  517  A.  80  sq.  rd.  by  10,000. 


COMPOUND   QUANTITIES.  157 

Surveyors'  Measure. 

311.  Surveyors  use  a  chain,  called  Gunter's  chain, 
which  is  4  rods,  or  66  feet,  long.  The  chain  has  100  links, 
and  therefore  links  are  written  as  hundredths  of  a  chain. 

Surveyors'  Table  of 


Measures  of  Length. 
7.92  in.        =  1  link  (1.). 
100  links    =  1  chain  (ch.) 

80  chains  =  1  mile  (mi.). 


Measures  of  Surface. 
16  sq.  rd.  =  1  sq.  ch. 
10  sq.  ch.  =  1  A. 
640  A.  =1  sq.  mi. 

1  sq.  mi.  =  1  section  (sec). 
36  sec.        =  1  township  (tp.). 


Exercise  76. 

1.  Reduce  10  ch.  to  inches. 

2.  Reduce  3168  in.  to  chains. 

3.  How  many  acres  are  there  in  a  township  ? 

4.  Reduce  6400  sq.  ch.  to  acres ;  to  square  miles. 

5.  Reduce  82,426  sq.  ch.  to  higher  units. 

6.  Add  4  sq.  mi.  412  A.  6  sq.  ch.  8  sq.  rd.;  7  sq.  mi. 
88  A.  2  sq.  ch.  11  sq.  rd.;  3  sq.  mi.  367  A.  7  sq.  ch.  2  sq.  rd.; 

11  sq.  mi.  344  A.  9  sq.  ch.  15  sq.  rd. 

7.  Subtract  1  mi.  75  ch.  85  1.  from  4  mi.  44  ch.  38  1. 

8.  What  is  the  area  of  a  field  if  it  can  be  divided  into 

12  lots  each  containing  2  sq.  ch.  7  sq.  rd.? 

.    9.    Multiply  3  sq.  mi.  172  A.  5  sq.  ch.  7  sq.  rd.  by  11. 

10.  Divide  6  sq.  mi.  422  A.  2  sq.  ch.  13  sq.  rd.  by '5. 

11.  A  field  is  divided  into  47  gardens  each  containing 
1  sq.  ch.  9  sq.  rd.     What  is  the  area  of  the  field  ? 

12.  A  field  containing  5  A.  4  sq.  ch.  11  sq.  rd.  is  divided 
into  25  equal  lots.     What  is  the  area  of  each  lot  ? 

13.  Find  the  rent  of  8  sq.  ch.  10  sq.  rd.  at  $2  an  acre. 

14.  If  a  field  contains  3  A.  6  sq.  ch.  12  sq.  rd.,  what  is 
it  worth  at  14  cents  a  square  foot  ? 


158  COMPOUND   QUANTITIES. 


Measures  of  Volume. 

312.  The  volume  of  a  body  is  the  number  of  cubic  units 
it  contains. 

Table. 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.). 

Note.     In  measures  of  volume,  the  scale  ascends  and  descends  by 
the  cubes  of  the  units  of  length  ;  thus  1728  =  12*  ;  27  =  38. 

313.  In  measuring  wood  and  small,  irregular  stones  the 
following  is  the 

Table. 

16  cubic  feet  =  1  cord  foot  (cd.  ft.). 

8  cord  feet,  or  >        ,        ,  ,   ,  . 
,OQ      ,  •    r  >  =  1  cord(cd.). 

128  cubic  feet      >  x      ' 

Note.     A  cord  is  a  pile  8  ft.  long,  4  ft.  wide,  and  4  ft.  high.     One 
foot  of  the  length  of  such  a  pile  is  called  a  cord  foot. 

Exercise  77. 

1.  Reduce  25  cu.  yd.  5  cu.  ft.  143  cu.  in.  to  cubic 
inches. 

2.  Reduce  921,730  cu.  in.  to  higher  units. 

3.  Wood  cut  in  lengths  of  4  ft.  is  piled  3£  ft.  high. 
How  long  must  the  pile  be  to  contain  2  cords  ? 

4.  How  many  cords  in  a  pile  of  4-ft.  wood  43  ft.  long 
and  6  ft.  high  ? 

5.  Add  130  cu.  yd.  5  cu.  ft.  820  cu.  in.;  56  cu.  yd.  20 
cu.  ft.  304  cu.  in.;  37  cu.  yd.  4  cu.  ft.  86  cu.  in.;  8  cu.  yd. 
10  cu.  ft.  129  cu.  in.;  12  cu.  yd.  19  cu.  ft.  175  cu.  in. 

6.  Subtract  32  cu.  yd.  13  cu.  ft.  1600  cu.  in.  from 
39  cu.  yd.  17  cu.  ft.  1400  cu.  in. 

7.  Multiply  12  cd.  4  cd.  ft.  by  14. 

8.  Divide  5  cu.  yd.  10  cu.  ft.  371  cu.  in.  by  6. 


COMPOUND   QUANTITIES.  159 

Measures  of  Value. 

314.  Money  is  the  measure  of  value. 

315.  Currency  is  the  medium  of  exchange  employed  in 
buying  and  selling. 

316.  Coins  or  Specie  are  stamped  pieces  of  metal  of 
fixed  purity  and  weight  issued  by  governments  as  money. 

317.  Bullion  is  uncoined  gold  or  silver,  of  standard 
purity,  usually  in  the  shape  of  bars. 

318.  Paper  Money  is  stamped  paper  containing  the 
promise  of  the  government  or  of  a  bank  of  issue  to  pay 
the  holder  on  presentation  a  specified  sum  of  standard 
coined  money. 

319.  United  States  Money.  The  unit  of  value  in  the 
United  States  is  the  dollar. 

Table. 
10~mills  (m.)    =  1  cent  (ct.). 
10  cents  =  1  dime  (d.). 

10  dimes  or  \       ,   .  „      .„,,. 

-iaa       *  (  =  1  dollar  ($)• 

100  cents        )  ' 

Cents  are  made  of  bronze;  half-dimes  of  nickel;  dimes,  quarter- 
dollars,  half-dollars,  and  dollars  of  silver;  pieces  of  two  and  a  half 
dollars,  five  dollars,  ten  dollars,  and  twenty  dollars  of  gold. 

A  10-dollar  gold  coin  is  called  an  eagle,  and  a  20-dollar  gold  coin 
is  called  a  double  eagle. 

Note.  The  standard  gold  dollar  weighs  25.8  gr.  and  contains 
23.22  gr.  of  pure  gold. 

320.  In  common  use  dimes  and  cents  are  read  together 
as  cents ;  figures  to  the  right  of  mills  are  read  as  the  deci- 
mal of  a  mill ;  mills  are  used  only  in  computation. 

Thus,  $4.27765  is  read  four  dollars  and  twenty-seven  cents,  seven 
and  sixty-five  hundredths  mills;  $4,273  is  reckoned  $4.27,  and  $4,277 
is  reckoned  $4.28. 


160  COMPOUND   QUANTITIES. 

321.  English  Money.  The  unit  of  value  in  Great 
Britain  is  the  pound  sterling,  which  is  equivalent  in  United 
States  money  to  $4.8665. 

Table. 

4  farthings  =  I  penny  (d.). 
12  pence  =  1  shilling  (s.). 
20  shillings  =  1  pound  (£). 

A  guinea  =  21s. ;  a  sovereign  =  20s. ;  a  half-sovereign  =  10s. ;  a 
crown  =  5s. ;  a  half-crown  =  2s.  6d. ;  a  florin  =  2s. 

Note  1.  The  penny  and  half-penny  are  made  of  copper  ;  the  three- 
penny piece,  the  six-penny  piece,  the  shilling,  the  florin,  the  half- 
crown,  and  the  crown  are  made  of  silver ;  the  half-sovereign  and  the 
sovereign  of  gold. 

Note  2.  Farthings  are  generally  written  as  the  common  fraction 
of  a  penny.  Thus,  1,  2,  and  3  farthings  are  written  £d.,  $d.,  and  |d., 
respectively. 

Exercise  78. 

1.  Eeduce  £583  6s.  8d.  to  pence. 

2.  Reduce  £79  18s.  ll£d.  to  farthings. 

3.  Reduce  28,572d.  to  higher  units. 

4.  Reduce  27,281  crowns  to  guineas. 

5.  Reduce  1,716,114  guineas  to  pounds. 

6.  Reduce  706,126d.  to  higher  units. 

7.  Add  £35  2s.  6fd.;  £18  5s.  4d.;  £27  3s.  10d.;  £12 
5d.;   £6  7s.  8d.;   £14  19s.  lid.;    £29  16s.  2d. 

8.  Subtract  £92  15s.  l£d.  from  £120  13s.  4d. 

9.  Multiply  £31  2s.  6^d.  by  8. 

10.  Divide  £394  2s.  10^d.  by  £5  2s.  4*d. 

11.  Divide  £108  15s.  4d.  by  13. 

12.  Find  the  value  in  United  States  money  of  the  money 
in  a  box  containing  35  sovereigns,  27  half-sovereigns,  13 
crowns,  41  half-crowns,  and  85  shillings. 


COMPOUND    QUANTITIES. 


161 


322.  Values  of  Foreign  Coins.     Oct.  1,  1897. 


Country. 

Standard. 

Monetary  Unit. 

Value  in 

terms  of 

U.  S.  gold 

dollar. 

Gold  and  silver.... 
Gold  .  .. 

Peso 

$0,965 

Crown 

0.203 

Gold  and  silver.... 

Silver 

Gold 

Franc  

0.193 

Bolivia 

Boliviano 

Milreis „ 

Dollar 

0.412 

Brazil 

0.546 

Gold.... 

1.000 

(except  Newfoundland). 
Central  Amer.  States  — 

Gold 

Gold 

Silver 

Colon  

0.465 

Dollar 

1.000 

Guatemala "] 

Honduras I 

Peso 

0.412 

Nicaragua f 

Salvador J 

Chile  

Gold 

Peso 

0.365 

Silver 

Silver 

0.664 

China 

0.608 
0.412 

Peso 

Gold  and  silver.... 
Gold 

Peso 

0.926 

Crown 

0.268 

Ecuador.... 

Silver 

Sucre 

0.412 

Egypt 

Gold 

Pound  (100  piasters)... 
Mark 

4.943 

Finland  ... 

Gold 

0.193 

France 

Gold  and  silver.... 

Gold 

Gold 

Franc  

0.193 

German  Empire ^. 

Great  Britain 

0.238 

Pound  sterling 

4.866^ 

Gold  and  silver 

Gold  and  silver  ... 
Silver 

0.193 

Haiti 

Gourde 

0.965 

India 

Rupee 

0.196 

Italy 

Gold  and  silver 

Gold 

Gold 

Lira                 

0.193 

Japan 

Yea    

0.498 

Liberia 

Dollar 

1.000 

Mexico , 

Silver. 

Dollar 

0.446 

Netherlands 

Gold  and  silver 

Gold 

0.402 

Newfoundland 

Dollar.... 

1.014 

Norway 

Gold    . 

0.268 

Persia 

Silver 

Kran 

0.076 

Peru 

Silver 

Sol ... 

0.412 

Portugal 

Gold    . 

Milreis 

1.080 

Russia 

Gold  

Gold  and  silver..... 

Gold 

Gold  and  silver....". 

Gold 

Gold  .. 

Ruble 

0.772 

Spain 

0.193 

Sweden 

0.268 

Switzerland 

Franc  

0.193 

Turkey 

Piaster  

0.044 

Uruguay 

1.034 

Venezuela 

Gold  and  silver.... 

0.193 

The  "  British  dollar  "  has  the  same  legal  value  as  the  Mexican  dollar  in  Hong- 
kong, the  Straits  Settlements,  and  Labuan. 

By  Imperial  ukase,  January  3/15, 1897,  l£  paper  rubles  =  1  gold  ruble,  giving 
paper  ruble  a  value  of  51 A  cents  U.  S.  money. 


162  COMPOUND   QUANTITIES. 

Measures  of  Time. 

323.  The  unit  of  time  is  the  day. 

324.  The  interval  of  time,  measured  from  the  instant 
the  sun  is  due  south  until  it  is  due  south  the  next  day,  is 
called  a  solar  day.  The  length  of  the  solar  day  varies 
slightly,  and  its  average  length,  called  the  mean  solar  day, 
is  the  unit  of  time. 

Table. 

60  seconds  (sec.)  =  1  minute  (min.). 

60  minutes  =  1  hour  (hr.). 

24  hours  =  1  day  (dy.). 

7  days  =  1  week  (wk.). 

366  days  =  1  common  year  (yr.). 

366  days  =  1  leap  year. 

100  years  =  1  century. 

325.  A  year  is  the  time  in  which  the  earth  performs  one 
revolution  round  the  sun,  and  consists  of  365.242218  mean 
solar  days. 

Note.  Before  the  time  of  Julius  Caesar  the  year  was  reckoned  as 
365  days.  On  the  supposition  that  365}  days  was  the  true  length,  he 
introduced  a  calendar  in  which  every  fourth  year  {every  year  which 
will  give  an  integral  quotient  when  its  number  is  divided  by  4)  was  to 
consist  of  366  days.     The  year  of  366  days  is  called  a  leap  year. 

The  error  of  the  Julian  calendar,  365.25  —  365.242219,  or  0.007781 
of  a  day,  would  amount  to  3.1124  days  in  four  centuries.  To  correct 
this  error  Pope  Gregory  XIII,  in  1582,  introduced  a  calendar  in  which 
three  leap  years  in  every  four  centuries  were  reckoned  as  common 
years.  Hence,  the  centuries  are  not  leap  years  unless  the  number  of 
the  century  divided  by  4  gives  an  integral  quotient.  Thus,  1600  and 
1884  were  leap  years ;  1800  and  1885  were  not ;  1900  will  not  be  a 
leap  year ;  2000  will  be  a  leap  year. 

The  present,  or  Gregorian,  calendar  leaves  a  slight  error  equal  to 
one  day  in  about  3300  years. 


dy. 

7. 

July 

.     31 

8. 

August  (Aug.)   .     . 

.     31 

9. 

September  (Sept.)  . 

.     30 

10. 

October  (Oct.)    .     . 

.     31 

11. 

November  (Nov.)    . 

.     30 

12. 

December  (Dec.)    . 

.     31 

COMPOUND   QUANTITIES.  163 

326.   The  year  is  divided  into  twelve  calendar  months. 

The  names  of  the  months  (mo.)  and  the  number  of  days  in  each  are  : 

dy. 

1.  January  (Jan.)    ...     31 

2.  February  (Feb.)       28  or  29 

3.  March  (Mar.)      ...     31 

4.  April  (Apr.)  ....     30 

5.  May 31 

6.  June 30 

February  has  28  days  in  common  years  and  29  days  in  leap  years. 

Note.     The  number  of  days  in  each  month  may  be  easily  remem- 
bered by  committing  to  memory  the  following  lines  : 

"  Thirty  days  hath  September, 
April,  June,  and  November  ; 
All  the  rest  have  thirty-one, 
Except  the  second  month  alone, 
Which  has  but  twenty-eight,  in  fine, 
Till  leap  year  gives  it  twenty-nine." 

A  lunar  month  is  the  time  between  two  new  moons,  and  is  a  little 
more  than  29  dy.  12  hr.  44  min. 


Exercise  79. 

1.  Reduce  6  hr.  17  min.  25  sec.  to  seconds. 

2.  Reduce  1  yr.  13  dy.  8  hr.  4  min.  to  minutes. 

3.  Reduce  48,567  min.  to  higher  units. 

4.  Reduce  7,423,922  sec.  to  higher  units. 

5.  How  many  minutes  are  there  from  midnight  of 
March  7  to  midnight  of  June  20  ? 

6.  Find  the  number  of  seconds  from  eight  o'clock  Mon- 
day morning  till  six  o'clock  the  next  Saturday  evening. 

7.  Which  of  the  years  1600,  1656,  1700,  1734,  1800, 
1818,  1880,  1900,  1924,  2000  are  leap  years  ? 

8.  Add  8  dy.  14  hr.  21  min.  37  sec;  44  dy.  17  hr.  13 
min.  32  sec;  208  dy.  9  hr.  47  min.  43  sec;  161  dy.  12  hr. 
53  min.  54  sec. ;  88  dy.  22  hr.  17  min.  50  sec. 


164  COMPOUND   QUANTITIES. 

9.  Subtract  2  yr.  213  dy.  17  tor.  48  niin.  48  sec.  from 
3  yr.  147  dy.  14  hr.  14  min.  32  sec. 

10.  Multiply  34  dy.  10  hr.  13  min.  12  sec.  by  108. 

11.  Divide  16  yr.  357  dy.  17  hr.  20  min.  48  sec.  by  18. 

12.  Divide  22  wk.  2  dy.  by  11  hr.  31  min.  12  sec. 

Difference  between  Two  Dates. 

327.    Examples.    1.  Find  the  difference  in  time  between 
July  4,  1897,  and  December  25,  1848. 

Solution.     In  finding  the  period  of  time  between  long  dates,  30 

yr.         mo.     dy.    days  are  considered  a  month.      As  July  is  the 

1897        7        4    seventh  and  December  the  twelfth  month,  we 

1848      12 26    write  7  and  12  instead  of  the  names  of  the  months. 

48        6        9    The  required  difference  is  48  yr.  6  mo.  9  dy. 

2.    Find  the  number  of  days  from  July  25  to  September  5. 

Solution.     The  number  of  days  in  July   =    6 

The  number  of  days  in  Aug.  =  31 

The  number  of  days  in  Sept.  =  _6 

Total  number  of  days  =  42 

In  finding  the  period  of  time  between  short  dates  the  exact  number 
of  days  is  counted  ;  and  the  last  day  named  is  included. 

Exercise  80. 

1.  Napoleon  was  born  Aug.  15,  1769,  and  died  at  the 
age  of  51  yr.  8  mo.  20  dy.  What  was  the  date  of  his 
death  ? 

2.  Daniel  Webster  was  born  Jan.  18,  1782,  and  died 
Oct.  24,  1852.     How  old  was  he  when  he  died  ? 

3.  A  note  dated  July  14,  1897,  has  63  days  to  run. 
When  is  the  note  due  ? 

4.  A  note  dated  Feb.  11,  1896,  has  93  days  to  run. 
When  is  the  note  due? 


COMPOUND   QUANTITIES. 


165 


5.  A  note  dated  Feb.  11,  1897,  has  63  days  to  run. 
When  is  the  note  due  ? 

6.  In  the  morning  of  July  5  a  man  went  into  the  coun- 
try for  his  vacation,  and  returned  in  the  evening  of  Sept.  26. 
Express  in  weeks  and  days  the  length  of  his  vacation. 

7.  Find  the  difference  in  time  between  Oct.  12,  1492, 
and  July  4,  1776. 

8.  Jan.  1,  1859,  fell  on  Saturday.  What  day  of  the 
week  was  Jan.  1,  1860  ?     Jan.  1,  1861  ? 

Circular  and  Angular  Measures. 

328.   Any   portion   of   the   circumference   of    a   circle 
is  called  an  arc. 


329.  If  a  straight  line  fixed  at  one  end  is  revolved  in  a 
plane,  the  other  end  describes  the  arc  of  a  circle  ;  and  the 
straight  line  in  moving  from  its  original  position  to  any 
other  given  position  describes  an  angle. 

Thus,  if  OA  revolves  on  a  fixed  point  0,  the  end  A  makes  the  cir- 
cumference ABC.     When  OA  has  reached  the  position  OB,  the  arc 

AB  has  been  made  by  A,  and  the  angle 
AOB  between  OA  and  OB  has  been 
made  by  OA.  The  angle  A  OB  is  such 
a  part  of  the  angular  magnitude  about 
0  as  AB  is  of  the  circumference. 

The  circumference  of  every  circle  is 
divided  into  360  equal  parts,  called 
degrees  (arc-degrees),  and  correspond- 
ing to  every  one  of  these  equal  parts 
is  an  angle  at  the  centre  of  the  circle. 
Hence,  the  whole  angular  magnitude 
about  any  point  in  a  plane  is  divided 
into  360  equal  parts  called  degrees  (angle-degrees),  and  the  number  of 
degrees  in  the  angle  formed  by  two  lines  drawn  from  the  centre  of  a 
circle  is  the  same  as  the  number  of  degrees  in  the  arc  which  is  inter- 
cepted between  these  two  lines. 


166  COMPOUND  QUANTITIES. 

330.  An  angle  described  by  a  line  making  one  fourth 
of  a  revolution  contains  90  degrees,  and  is  called  a  right 
angle,  as  AOB  ;  and  OA  and  OB  are  said  to  be  perpe?idicu- 
lar  to  each  other.  An  angle  less  than  a  right  angle  is 
called  an  acute  angle;  an  angle  greater  than  a  right  angle 
and  less  than  two  right  angles  is  called  an  obtuse  angle. 

Table. 

60  seconds  (")  =  1  minute  C). 
60  minutes       =  1  degree  (°). 
360  degrees        =  1  revolution  or  circumference. 

Note.  A  degree  of  the  circumference  of  the  earth  at  the  equator 
contains  60  geographical  miles,  or  69.16  statute  miles. 


Exercise  81. 

1.  Reduce  2°30'25"  to  seconds. 

2.  Keduce  15°  3' 22"  to  seconds. 

8.  Reduce  56,760"  to  higher  units. 

4.  Reduce  212,221"  to  higher  units. 

5.  Add  60°  50'  50";  20°  41' 52";  30°  25' 20";  20°  32' 43" 

6.  Subtract  58°  33'  36"  from  90°  11'  21". 

7.  Multiply  12°  14' 32"  by  48. 

8.  Divide  321°  49' 24"  by  22. 

9.  Divide  38°  37'  42"  by  5°  31'  6". 


331. 


Miscellaneous  Tables. 


Numbers. 


12  units  =  1  dozen. 
12  dozen  =  1  gross. 
12  gross  =  1  great  gross. 
20  units  =  1  score. 


Papeb. 


24  sheets     =  1  quire. 
20  quires    =  1  ream. 

2  reams     =  1  bundle. 

6  bundles  =  1  bale. 


COMPOUND   QUANTITIES. 


167 


Weights. 


1  bu.  of  corn  or  rye  =  56  lb 

1  bu.  of  corn  meal,  rye  }  _ 

meal,  or  cracked  corn 
1  bu.  of  wheat 


\ 


501b. 
=  60  lb. 


1  bu.  of  potatoes,  beets,  )  _ 

etc.  } 

1  bu.  of  beans  or  peas     =  60  lb. 
1  bu.  of  oats  =  32  lb. 


=  48  lb. 
451b. 


1  bu.  of  barley 

1  bu.  of  timothy-  )  _ 

seed  >  " 

1  stone  of  iron  or  >  __ 

lead  > 

1  pig  of  iron  or  lead  =  21|  stone. 
1  fother  of  iron  or  )       _     . 

lead  |  =  8^ 


lb. 


The  weight  of  a  bushel  of  barley,  oats,  etc. ,  varies  slightly  in  differ- 
ent States,  but  the  weights  here  given  are  those  generally  adopted  in 
business  transactions. 


Weights. 

1  bbl.  of  flour  =  196  lb. 

1  bbl.  of  pork  or  beef  =  200  lb. 
1  cask  of  lime  ==  240  lb. 

1  cental  of  grain  ,        =  100  lb. 
1  quintal  of  fish   "       =  100  lb. 


Books. 

A  book  of  sheets  folded  in  : 
2  leaves  is  a  folio. 
4  leaves  is  a  quarto. 
8  leaves  is  an  octavo. 

12  leaves  is  a  duodecimo. 

16  leaves  is  a  16mo. 


.  Denominate  Fractions. 

332.   Examples.     1.    Express  §  rd.  in  yards,  feet,  and 

inches. 

Solution.     §  rd.  =  §  of  b\  yd.  =  3§  yd. 
2  y(j.  =  2  0f  3  ft.  -  2  ft. 

Hence,  f  rd.  =  3  yd.  2  ft. 


2.   Find  the  value  of  }  of  £3  2s.  4d. 


8[3_ 


J_  Solution.     We  divide  by  8  to  get  £  of  £3  2s. 

9|  4d.  and  multiply  the  quotient  by  3  to  get  f  of 

_3_  £3  2s.  4d.     Hence,  |  of  £3  2s.  4d.  is  £1  3s.  4£d. 

H 


168 


COMPOUND   QUANTITIES. 


3.   Find  the  value  of  0.3975  of  a  mile. 


0.3975 
320 
79500 

11925 

127.2 
16* 
3.3 


12 
3.6 


Solution.     0.3975  mi.  =  0.3975  of  320  rd. 
=  127.2  rd. 
0.2  rd.  =  0.2  of  16 £  ft.  =  3.3  ft. 
0.3  ft.  =  0.3  of  12  in.  =  3.6  in. 
Hence,  0.3975  mi.  =  127  rd.  3  ft.  3.6  in. 


Exercise  82. 


Find  the  value  of : 

1.  $  of  a  mile. 

2.  T\  of  an  acre. 

3.  |  of  a  hundred  weight. 

4.  §  of  a  pound  sterling. 
5. 
6. 


ft  of  a  mile. 
^T  of  an  acre. 


7.  |  of  a  degree. 

8.  £  of  a  year. 

9.  0.15625  of  a  bushel. 


10.  0.625  of  a  gallon. 

11.  0.875  of  a  leap  year. 

12.  0.325  of  a  pound  troy. 

13.  6 £  of  3  A.  101 J  sq.  rd. 

14.  l^of  7  hr.  21  min.  27  sec. 

15.  10.0175  of  1  dy.  13  hr. 

16.  17^  of  10  yd.  2  ft.  3£  in. 

17.  0.01284  of  14  mi. 

18.  0.42776  of  12  t.  10  cwt. 


Find  the  value  of  : 

19.  f  of  1  lb.  +  3$  oz.  +  5§  dwt. 

20.  0.35  of  4  lb.  5  oz.  6  dwt.  16  gr. 

21.  3.726  mi. -33.57  rd. 

22.  T33  of  a  year  -}-  ^  of  a  week  -f-  t7*  °f  an  hour. 

23.  5.268  of  2  dy.  +  2.829  of  16  hr.  +  0.9528  of  25  min. 

24.  ^  of  a  mile  +  f  of  40  rd.  -f  f  of  a  yard. 

25.  f  ,of  2  cwt.  84  lb.  +  f  of  5  cwt.  98  lb.  +  £  of  7£  lb. 

26.  f  of  21  ft.  7  in.  +  0.855  of  16  ft.  2  in.  +  0.365  of  1  ft. 

27.  0.9  of  4  A.  17  sq.  rd.  -  \\  of  3  A.  15  sq.  rd. 

28.  0.652  of  2  cu.  yd.  7  cu.  ft.  —  0.888  of  1  cu.  yd.  2  cu.  ft. 

29.  0.456  of  12  bu.  3  pk.  —  0.654  of  5  bu.  2  pk. 


COMPOUND   QUANTITIES.  169 

333.    Examples.     1.    Express  10  hr.  33  min.  36  sec.  as 
the  fraction  of  a  day. 

Solution.     36  sec.  =  f  #  min.  =  f  min. 
33f  min.  =  ^  hr.  =  iff  hr.  =  if  hr. 

1014 

10tt  hr.  =  ^f  dy.  =  f  ft  dy.  =  H  dy. 
Hence,  10  hr.  33  min.  36  sec.  =  \\  dy. 

2.    Express  127  rd.  0  ft.  7.92  in.  as  the  decimal  of  a  mile. 
12 


161 
320 


7.92    in.        Solution.        7.92  -fl2    =0.66. 

0.66    ft.  0.66  -M6|  =  0.04. 

127.04    rd.  127.04  +  320  =  0.397. 


0.397  mi.       Hence,  127  rd.  0  ft.  7.92  in.  =  0.397  mi. 


3.   Express  1  yd.  2  ft.  4  in.  as  the  common  fraction,  and 
as  the  decimal,  of  5  yd.  1  ft.  4  in. 

Solution.     1  yd.  2  ft.  4  in.  =  64  in. ;  5  yd.  1  ft.  4  in.  =  196  in. 
-64  in.  _  64  _  16         _„, 

Hence,  1  yd.  2  ft.  4  in.  is  £f,  or  0.32653+  of  5  yd.  1  ft.  4  in. 

Exercise  83. 
Express  : 

1 .  A  pound  avoirdupois  as  the  fraction  of  a  pound  troy. 

2.  An  ounce  avoirdupois  as  the  fraction  of  an  ounce 
troy. 

3.  363  sq.  yd.  as  the  fraction  of  an  acre. 

4.  f  of  £2  Is.  3d.  +  T5T  of  £1  4s.  9d.  as  the  fraction  of 
£2  14s. 

5.  2  mi.  138  rd.  1  yd.  as  the  fraction  of  3  mi.  265  rd. 
3  yd.  1  ft.  6  in. 

6.  f  of  560  lb.  as  the  fraction  of  5  long  tons. 

7.  §  of  200  rd.  as  the  fraction  of  4  mi. 

8.  tf  of  2  dy.  2  hr.  24  min.  as  the  fraction  of  2  wk.  1  dy. 


170  COMPOUND   QUANTITIES. 

9.   $  of  the  difference  between  3  yd.  2  ft.  11  in.  and  10 
yd.  7  in.  as  the  fraction  of  8  yd. 

10.  £f  of  the  difference  between  |  of  7  hr.  and  ^  of  15 
min.  as  the  fraction  of  12  hr.  18  min. 

11.  £  pt.  as  the  fraction  of  a  gallon. 

12.  16s.  3|d.  as  the  decimal  of  a  pound. 

13.  233  rd.  9  ft.  10.8  in.  as  the  decimal  of  a  mile. 

14.  71  sq.  rd.  54  sq.  ft.  64.8  sq.  in.  as  the  decimal  of  an 
acre. 

15.  15  hr.  14  min.  6  sec.  as  the  decimal  of  2  days. 

16.  38  sq.  rd.  21  sq.  yd.  5  sq.  ft.  108  sq.  in.  as  the  deci- 
mal of  an  acre. 

17.  3  mi.  242  rd.  2  yd.  2  ft.  3  in.  as  the  decimal  of  7 
mi.  160  rd. 

18.  5  hr.  13  min.  30  sec.  as  the  decimal  of  a  week. 

19.  27°  14' 45"  as  the  decimal  of  90°. 

20.  54  dy.  2  hr.  40  min.  as  the  decimal  of  365 J  days. 

21.  2  lb.  avoirdupois  as  the  decimal  of  10  lb.  troy. 

22.  44,920.9025  hr.  as  the  decimal  of  a  year. 

23.  14.52  sq.  yd.  as  the  decimal  of  a  square  chain. 

24.  8  cwt.  77  lb.  9.6  oz.  as  the  decimal  of  a  ton. 

25.  What  part  of  4  lb.  1  oz.  8  dwt.  15  gr.  is  1  lb.  1  oz. 
9  dwt.  15  gr.? 

26.  What  part  of  2  mi.  is  §  of  6  rd.  3  yd.  2  in.? 

27.  What  part  of  a  bushel  is  1  pk.  2  qt.  1  pt.? 

28.  What  part  of  20  acres  is  19  A.  3.5  sq.  ch.? 

29.  What  part  of  5  tons  is  3  t.  240  lb.? 

30.  What   part   of  an  acre   is  38  sq.   rd.   194  sq.   ft. 
108  sq.  in.? 

31.  Express  2  lb.  9  oz.  21  dwt.  as  the  decimal  of  4  lb. 
7  oz.  19  dwt. 

32.  Express  17  wk.  6  dy.  22  hr.  39  min.  as  the  decimal 
of  35  wk.  3  dy.  15  hr.  25  min. 

33.  What  part  of  61  ft.  3  in.  is  8  ft.  7  in,? 


COMPOUND   QUANTITIES.  171 


Longitude  and  Time. 

334.  Meridians  are  imaginary  lines  drawn  straight 
around  the  earth  through  both  poles. 

335.  Longitude  is  reckoned  in  degrees,  minutes,  and 
seconds  east  or  west  from  a  standard  meridian,  as  the 
meridian  of  Greenwich,  near  London.  Since  longitude 
is  reckoned  east  and  west  from  a  given  meridian  the 
longitude  of  a  place  is  never  greater  than  180°,  half  the 
distance  round  the  earth. 

336.  When  two  places  are  both  east  or  both  west  of 
the  standard  meridian,  the  difference  of  their  longitudes 
is  found  by  subtracting  the  one  from  the  other. 

When  one  place  is  east  and  the  other  west  of  the 
standard  meridian,  the  difference  of  their  longitudes  is 
found  by  adding  the  two  longitudes. 

If  the  sum  Of  the  two  longitudes  is  greater  than  180°, 
this  sum  must  be  subtracted  from  360°  to  obtain  the  cor- 
rect difference  of  longitude. 

Thus,  if  one  longitude  is  130°  west  and  another  120°  east,  the  dif- 
ference of  their  longitudes  is  360°  -  (130°  +  120°),  or  110°. 

337.  As  the  earth  turns  upon  its  axis  once  in  twenty- 
four  hours,  a  point  on  the  earth's  surface  will  describe 
a  circumference  (360°)  in  twenty-four  hours.  Therefore, 
longitude  may  be  reckoned  in  time  as  well  as  in  degrees. 

In  one  hour  a  point  on  the  earth's  surface  describes  -fa 
of  360°  =  15°;  in  one  minute,  ^  of  15°=  15';  and  in  one 
second,  ^o  of  15'=  15". 

Again,  since  it  requires  one  hour  (60  min.)  for  a  point  to 
pass  over  15°,  to  pass  over  1°  it  requires  TV  of  60  min.  = 
4  min.;  and  to  pass  over  1'  it  requires  ^  of  4  min.  = 
4  sec. 


172  COMPOUND   QUANTITIES. 

338.  Examples.  1.  Express  20°  36f  15"  of  longitude 
in  time. 

Solution.  Since  15°  longitude  give  1  hr.  in  time,  15'  longitude 
1  min.,  and  15"  longitude  1  sec,  divide 

15)20°     36/      15"  20°  36'  16"  by  15,  as  in  compound  divi- 

1  hr.  22  min.  25  sec.  sion,  and  the  quotient  will  be  the  time 
required,  1  hr.  22  min.  25  sec. 

2.   Express  1  hr.  4  min.  4  sec.  in  degrees. 

Solution.     Since  1  hr.  of  time  equals  15°  of 
1  hr.  4  min.  4  sec.     longitude,  1  min.  of  time  15',  and  1  sec.  of  time 

15  15",  multiply  1  hr.  4  min.  4  sec.  by  15,  as  in 

16°     r     0"  compound  multiplication,  and  the  product  will 

be  the  longitude  required.     Hence, 

339.  If  longitude  is  expressed  in  degree-measures, 
divide  by  15 ;  the  quotient  gives  the  longitude  in  time- 
measures. 

If  longitude  is  expressed  in  time-measures,  multiply  by 
15 ;  the  product  gives  the  longitude  in  degree-measures. 

Exercise  84. 

Find  the  difference  in  longitude  between  two  cities,  if 
the  difference  in  time  is  : 

1.  1  hr.  15  min.  5.  6  hr.  12  min.  30  sec. 

2.  2  hr.  11  min.  6.  4  hr.  8  min.  12  sec. 

3.  5  hr.  10  min.  10  sec.  7.  18  hr.  10  min. 

4.  3  hr.  25  min.  35  sec.  8.  15  hr.  15  min.  15  sec. 

Find  the  difference  in  time  between  two  cities,  if  the 
difference  in  longitude  is  : 

9.  9°  20'.  13.  120°  14' 30". 

10.  70°  30'.  14.  100°  45' 54". 

11.  56°  36' 12".  15.  2°  2' 2". 

12.  108°  32' 36".  16.  75°  10'. 


COMPOUND    QUANTITIES.  173 

17.  Find  the  difference  in  time  between  New  York, 
longitude  74°  0'  3"  West,  and  San  Francisco,  longitude 
122°  26'  15"  West. 

18.  The  difference  in  time  between  Berlin  and  New  York 
is  5  hr.  49  min.  35  sec.  What  is  the  difference  in  longi- 
tude? 

340.  Since  the  sun  appears  to  move  from  east  to  west, 
sunrise  will' occur  earlier  at  all  points  east  and  later  at  all 
points  west  of  a  given  place.  Hence,  clock-time  will  be 
later  in  all  places  east  and  earlier  in  all  places  west  of  a 
given  meridian. 

Therefore,  if  the  time  of  a  place  is  given, 

To  find  the  time  of  a  place  east,  add  to  the  given  time 
the  difference  of  time  between  the  two  places. 

To  find  the  time  of  a  place  west,  subtract  from  the  given 
time  the  difference  of  time  between  the  two  places. 

341.  To  Find  the  Difference  in  Clock-time  when  the 
Difference  in  Longitude  is  Known. 

When  it  is  noon  at  Boston  (long.  71°  3'  30"  West),  what 
is  the  time  at  Paris  (long.  2°  20'  22"  East)  ? 

71°     8'    30"  W. 

2°    20'    22"  E. 


23'    52".  .  .  difference  in  longitude. 


15)73°    23'    52' 


4  hr.  53  min.  35T7?  sec.  difference  in  time. 

Solution.  Since  Boston  is  west  and  Paris  is  east  of  the  meridian 
of  Greenwich,  the  difference  between  their  longitudes  is  found  by 
taking  the  sum  of  their  longitudes. 

The  difference  between  their  longitudes,  73°  23'  52",  is  equivalent 
to  4  hr.  53  min.  35^  sec,  and  as  Paris  is  east  of  Boston,  the  time  at 
Paris  is  found  by  adding  the  4  hr.  53  min.  35T^  sec.  to  the  time  at 
Boston,  making  53  min.  35T75  sec.  past  4  p.m. 


174  COMPOUND   QUANTITIES. 

Exercise  85. 
The  longitude  of  some  public  building  in  : 

(1)  Berlin  is  13°  23'  43"  E.  (7)  Jerusalem,  35°  32'  E. 

(2)  Rome,  12°  27'  14"  E.  (8)  Bombay,  72°  54'  E. 

(3)  Constantinople,  28°  59'  E.  (9)  Calcutta,  88°  19'  2"  E. 

(4)  Pekin,  116°  23'  45"  E.  (10)  Chicago,  87°  35'  W. 

(5)  San  Francisco,  122°  26'  15"  W.  (11)  New  York,  74°  0'  3"  W. 

(6)  St.  Louis,  90°  15'  15"  W.  (12)  Montreal,  73°  25'  W. 

What  is  the  clock-time  at  each  of  the  above  cities : 

1.  When  it  is  noon  at  Greenwich  ? 

2.  When  it  is  half-past  four  p.m.  at  Chicago  ? 

3.  When  it  is  eight  o'clock  a.m.  at  Constantinople  ? 

When  it  is  noon  at  Greenwich  the  time  at : 

(1)  Boston,  Mass.,  is  7  hr.  15  min.  46  sec.  a.m. 

(2)  Columbia,  S.  C,  6  hr.  35  min.  32  sec.  a.m. 

(3)  Salt  Lake,  4  hr.  30  min.  a.m. 

(4)  Albany,  N.  Y.,  7  hr.  5  min.  1  sec.  a.m. 

(5)  Harrisburg,  Penn.,  0  hr.  52  min.  40  sec.  a.m. 

(6)  New  Orleans,  La.,  6  hr.  a.m. 

(7)  Columbus,  O.,  6  hr.  27  min.  48  sec.  a.m. 

(8)  Washington,  D.  C,  6  hr.  51  min.  44  sec.  a.m. 

(9)  Springfield,  111.,  6  hr.  1  min.  48  sec.  a.m. 

4.  What  is  the  longitude  of  each  of  the  above  cities  ? 

Note.  Standard  time  is  the  clock-time  of  some  selected  meridian. 
Eastern  standard  time  is  the  clock-time  of  the  meridian  75°  west  of 
Greenwich,  and  is  five  hours  slower  than  Greenwich  time.  Central 
standard  time  is  the  clock-time  of  90°  west  of  Greenwich,  and  is  just 
one  hour  slower  than  Eastern  standard  time.  Mountain  standard  time 
is  the  clock-time  of  the  meridian  of  105°,  and  is  one  hour  slower  than 
that  of  90°.  Western  standard  time  is  the  clock-time  of  the  meridian 
of  120°,  and  is  one  hour  slower  than  that  of  105°.  The  railroads  and 
many  cities  and  towns  of  the  United  States  have  adopted  standard  time. 

Places  not  more  than  7|°  east  or  west  of  the  meridians  of  75°,  90°, 
105°,  120°  are  reckoned  to  have  the  same  time  respectively  as  places 
on  these  meridians. 


COMPOUND   QUANTITIES.  175 

Exercise  86. — Eeview. 

1.  Reduce  7  gal.  3  qt.  1  pt.  to  gallons  and  the  decimal 
of  a  gallon. 

2.  Reduce  £4.375  to  pounds,  shillings,  and  pence. 

3.  Reduce  7.6875  gal.  to  gallons,  quarts,  and  pints. 

4.  If  $4.85  is  equal  to  a  pound,  reduce  to  pounds,  shil- 
lings, and  pence  $5,875  ;  $7.38  ;  $17.85;  $21.75. 

5.  How  many  square  yards  in  3.7156  A.? 

6.  If  2  qt.  of  linseed  oil  are  mixed  with  \  pt.  spirits  of 
turpentine,  what  fraction  of  the  mixture  is  turpentine  ? 
How  much  turpentine  in  one  pint  of  the  mixture  ? 

7.  Reduce  5.1732  mi.  to  yards,  feet,  and  inches. 

8.  If  a  man  walks  88  mi.  in  26  hr.,  how  many  feet  does 
he  walk  in  a  second  ? 

9.  Of  a  mixture  of  sand  and  lime  0.27  of  the  weight  is 
lime.  How  many  ounces  of  lime  in  a  pound  of  the  mix- 
ture ?  How  many  troy  grains  of  lime  in  an  avoirdupois 
pound  of  the  mixture  ? 

10.  A  gill  of  water  is  put  into  a  quart  measure,  and  the 
measure  then  filled  with  milk.  What  part  of  the  mixture 
is  water  ? 

11.  Reduce  555  ft.  to  the  decimal  of  a  mile. 

12.  Reduce  1  mi.  13  rd.  2  yd.  2  ft.  6  in.  to  inches. 

13.  How  many  cubic  inches  in  2 \  cu.  ft.  ? 

14.  How  many  pounds  avoirdupois  does  a  cubic  yard  of 
water  weigh  if  a  cubic  foot  weighs  1000  oz.  ? 

15.  Express  the  weight  of  a  cubic  yard  of  water  as  the 
decimal  of  a  ton. 

16.  What  is  the  weight  of  7  bu.  3^  pk.  of  potatoes  ? 

17.  A  farmer  sowed  5  bu.  1  pk.  1  qt.  of  seed,  and  har- 
vested from  it  103  bu.  3  pk.  5  qt.  How  much  did  he 
raise  from  a  bushel  of  seed  ? 

18.  How  many  bushels  in  5  tons  of  oats  ? 


176  COMPOUND   QUANTITIES. 

19.  How  many  bottles,  each  holding  1  pt.  3  gi.,  can  be 
filled  from  a  barrel  of  cider  ? 

20.  If  a  steamer  makes  13  mi  6  rd.  an  hour,  how  far 
will  she  go  between  6  a.m.  and  6  p.m.?  How  many  hours 
will  she  require  to  make  113  mi.  ? 

21.  If  a  train  runs  at  the  average  rate  of  111  rd.  a 
minute,  how  many  hours  will  it  require  to  run  from  Boston 
to  Buffalo,  498  mi.  ? 

22.  What  is  the  cost  of  12  A.  146  sq.  rd.  of  land  at 
$16.25  an  acre  ? 

23.  What  is  the  cost  of  8  t.  3  cwt.  27  lb.  of  coal  at  $5.75 
a  ton? 

24.  What  is  the  cost  of  7  t.  1560  lb.  of  hay  at  $15.50  a 
ton? 

25.  What  is  the  cost  of  a  car  load  of  wheat  weighing 
20,000  lb.  at  $1.05  a  bushel  ? 

26.  Reduce  5  rd.  4  yd.  2 J  ft.  to  the  decimal  of  a 
mile. 

27.  Reduce  9  sq.  ch.  11.25  sq.  rd.  to  the  decimal  of  an 
acre. 

28.  Reduce  0.09375  bu.  to  quarts. 

29.  Reduce  7560  chains  to  miles. 

30.  How  many  gross  are  2000  pens  ? 

31.  Find  the  cost  of  27.248  A.  at  $93.75  an  acre. 

32.  Which  is  the  greater,  2.8  of  3  ft.  11  in.  or  3.11  of  2 
ft.  8  in.,  and  by  how  much  ? 

33.  Reduce  171  lb.  6  oz.  troy  to  the  decimal  of  a  ton 
avoirdupois. 

84.  Express  14.52  sq.  yd.  as  the  decimal  of  a  square 
chain. 

35.  If  a  sovereign  is  equal  to  25.22  francs,  or  to  $4.85, 
what  decimal  of  a  dollar  is  a  franc  ? 

36.  If  0.327  of  some  work  is  done  in  3  hr.  38  min.,  how 
long  will  the  whole  work  require  ? 


COMPOUND   QUANTITIES.  177 

37.  A  can  run  a  mile  in  7.68  min.;  B  can  run  at  the 
rate  of  7.68  mi.  an  hour.     Which  is  the  faster  runner  ? 

38.  How  many  miles  an  hour  does  a  person  walk  who 
takes  2  steps  a  second  and  1900  steps  in  a  mile  ? 

39.  If  an  ounce  troy  of  gold  is  worth  $20,  what  is  the 
value  of  a  pound  avoirdupois  ? 

40.  Two  stars  cross  the  meridian  at  6  hr.  4  min.  42.3 
sec.  and  7  hr.  2  min.  57.21  sec,  respectively.  What  is  the 
interval  between  the  observations  ? 

41.  How  long  will  it  take  to  fill  %%  of  a  cistern,  when 
the  whole  requires  6  hr.  10  min.? 

42.  The  circumference  of  a  circle  is  6  yd.  1  ft.  5.1  in. 
What  is  the  length  of  an  arc  of  55°  ? 

43.  Multiply  2  t.  16  cwt.  63f  lb.  by  If. 

44.  Into  how  many  shares  has  £120  been  divided  when 
each  share  is  £3  8s.  6f  d.? 

45.  If  ||  of  one  line  is  equal  to  §  of  another  line,  which 
is  the  greater  tm    What  fraction  of  the  greater  is  the  less  ? 

46.  Multiply  5  mi.  206  rd.  2  ft.  2  in.  by  786. 

47.  The  returns  of  a  gold  mine  are  241 1.  of  ore  yielding 

2  oz.  1  dwt.  15  gr.  of  fine  gold  a  ton,  and  193"  t.  yielding 
1  oz.  12  dwt.  9  gr.  a  ton.  Find  the  value  of  the  whole 
yield,  at  $19.45  an  ounce. 

48.  Divide  93  long  tons  56  lb.  by  23  lb.  5  oz. 

49.  Telegraph  poles  on  railroads  are  generally  erected 
at  intervals  of  88  yd.  Show  that  if  a  passenger  counts 
the  number  of  poles  which  the  train  passes  in  three 
minutes,  that  number  will  express  the  number  of  miles  an 
hour  the  train  is  going. 

50.  If  Greenwich  time  is  5  hr.  8  min.  16  sec.  later 
than  Washington  time,  and  Chicago  is  87°  35'  W.,  what  is 
the  difference  between  Washington  time  and  Chicago  time  ? 

5 1 .  What  fraction  of  21  cu.  yd.  11  cu.  ft.  1215  cu.  in.  is 

3  cu.  yd.  1  cu.  ft.  1161  cu.  in.? 


178  COMPOUND   QUANTITIES. 

52.  How  many  minutes  in  the  first  three  months  of 
1895  ?     How  many  in  the  first  three  months  of  ^896  ? 

53.  A  knot  is  ^  of  a  degree,  and  a  mile  is  0.01477  of  a 
degree.     Find  in  miles  the  value  of  a  knot  to  five  decimals. 

54.  The  captain  of  a  steamer,  sailing  from  Liverpool, 
found  on  taking  an  observation  that  the  sun  crossed  his 
meridian  at  42  min.  5  sec.  past  one  o'clock  p.m.  by  Green- 
wich time.     Find  his  longitude. 

55.  If  a  walk  6  ft.  wide  is  made  round  a  park  600  ft. 
square  within  the  enclosure,  how  many  square  yards  will 
the  walk  contain  ? 

56.  How  many  pickets  3  in.  wide,  placed  3  in.  apart, 
will  be  required  to  fence  a  rectangular  lot  231  ft.  long  and 
99  ft.  wide  ?     What  will  they  cost  at  $3.25  per  hundred  ? 

57.  The  length  of  a  year  is  365.242218  mean  solar  days. 
Express  the  length  of  a  year  in  days,  hours,  minutes,  and 
seconds. 

58.  The  Flying  Dutchman  Express  runs  from  London  to 
Exeter,  a  distance  of  193£  mi.,  in  4J  hr.,  making  one  stop 
of  10  min.,  two  of  5  min.  each,  and  one  of  3  min.  What 
is  its  average  speed  per  hour  when  in  motion  ? 

59.  The  Scotch  Express  runs  from  London  to  Edin- 
burgh, a  distance  of  393f  mi.,  in  9  hr.,  making  one  stop  of 
30  min.,  three  of  5  min.  each,  and  one  of  3  min.  What  is 
its  average  speed  per  hour  when  in  motion  ? 

60.  The  Empire  State  Express  runs  from  New  York  to 
Buffalo,  a  distance  of  439  mi.,  in  8  hr.  15  min.,  making  two 
stops  of  3  min.  each  and  two  stops  of  2  min.  each.  What 
is  its  average  speed  per  hour  when  in  motion  ? 

61.  How  many  dollars  worth  4s.  2d.  each  will  pay  a 
bill  of  £11  17s.  6d.? 

62.  The  lunar  month  is  29.53059  days.  Express  the 
length  of  a  lunar  month  in  days,  hours,  minutes,  and 
seconds. 


.  CHAPTER   IX. 

PROBLEMS. 

342.  Arithmetical  Analysis.  If  the  value  of  any 
number  of  units  is  given,  we  may  by  division  find  the 
value  of  one  unit  of  the  same  kind,  and  by  multiplication 
the  value  of  any  number  of  units  of  the  same  kind.  The 
solution  which  combines  these  two  processes  is  called 
analysis. 

Example.  If  18  yards  of  cloth  cost  $45,  what  will  be 
the  cost  of  27  yards  of  cloth  ? 

Solution.  If  18  yd.  of  cloth  cost  $45,  1  yd.  will  cost  T\  of  .$45, 
or  $2|.     If  1  yd.  of  cloth  costs  $2£,  27  yd.  will  cost  27  X  $2£,  or  $67£. 

Exercise  87. 

1.  If  15  yards  of  silk  cost  $18.75,  what  will  be  the  cost 
of  20£  yards  ? 

2.  If  3f  pounds  of  tea  cost  $3.80,  how  many  pounds  can 
be  bought  for  $21.89? 

3.  If  T\  of  a  ton  of  coal  costs  $1.12,  what  is  the  price 
of  5£  cwt.? 

4.  If  T2T  of  a  piece  of  work  is  done  in  25  days,  what 
fraction  of  the  work  will  be  done  in  llf  days  ? 

5.  A  bankrupt's  debts  are  $2520,  and  the  value  of  his 
property  is  $1890.     How  much  can  he  pay  on  a  dollar  ? 

6.  If  a  bankrupt's  debts  are  $4264,  and  he  pays  62£ 
cents  on  a  dollar,  what  are  his  assets  ? 


180  PROBLEMS. 

7.  If  an  ounce  of  gold  is  worth  $20  67,  what  is  the 
value  of  0.04  of  a  pound  ? 

8.  A  man  spent  £  of  his  money  for  dry  goods,  \  of  the 
remainder  for  groceries,  and  had  $15  left.  How  much  had 
he  at  first  ? 

Solution.  After  spending  f  of  his  money  he  had  \  left.  After 
spending  \  of  f  of  his  money  he  had  left  \  of  f  =  /5.  Then,  $15  =  & 
of  the  money  he  had  at  first. 

9.  Sampson  &  Reed  sold  f  of  a  lot  of  wheat  to  one 
man,  f  of  the  remainder  to  another,  and  had  93  bushels 
left.     How  much  had  they  at  first  ? 

10.  In  a  certain  school  ^  of  the  scholars  are  girls  ;  \  of 
the  boys  are  over  16  years  old,  and  6  boys  are  under  16. 
How  many  girls  and  how  many  scholars  are  there  in  the 
school  ? 

11.  In  a  certain  school  \\  of  the  scholars  are  boys; 
fa  of  the  girls  are  under  16,  and  13  girls  are  over  16.  How 
many  boys  and  how  many  girls  are  there  in  the  school  ? 

12.  If  from  a  certain  number  f  of  it  is  subtracted,  then  £ 
of  the  remainder,  then  }  of  that  remainder,  6  still  remains. 
What  is  the  number  ? 

13.  A  ship's  cargo  sold  for  $45,000  belongs  to  three 
partners.  A  owns  $  of  $  of  it,  B's  share  is  equal  to  3^ 
of  |  of  A's  share,  and  C  owns  the  remainder.  What  does 
each  receive  from  the  sale  ? 

14.  A  man  bequeathed  T\  of  his  property  to  A,  £  of  it 
to  B,  £  to  C,  i  to  D,  and  the  remainder,  $550,  to  E.  What 
was  the  value  of  his  whole  property  ? 

15.  A  farmer  raised  321  bu.  3  pk.  of  corn  from  9  acres 
of  land.  At  the  same  rate,  what  would  be  the  yield  from 
25  acres  ? 

16.  If  7  horses  eat  21  bushels  of  oats  in  16  days,  how 
many  days  will  99  bu.  3  pk.  last  them  ? 


PROBLEMS.  181 

17.    If  12  horses  can  plow  96  acres  in  6  days,  how  many 
horses  will  plow  64  acres  in  8  days  ? 

Solution.     In  6  days  96  acres  can  be  plowed  by  12  horses. 
In  1  day  96  acres  can  be  plowed  by  6  X  12  horses. 

In  1  day  1  acre  can  be  plowed  by  -tt —  horses. 

9o 

6  X  12 

In  8  days  1  acre  can  be  plowed  by —  horses. 

o  X  \jo 

In  8  days  64  acres  can  be  plowed  by  —  —  horses. 

o  X  9o 


18.  If  40  acres  of  grass  is  mowed  by  8  men  in  7  days, 
how  many  acres  will  be  mowed  by  24  men  in  28  days  ? 

Solution.  24  men  will  mow  three  times  as  much  as  8  men  in  the 
same  time  ;  the  same  number  of  men  will  mow  four  times  as  much  in 
28  days  as  in  7  days.  Hence,  24  men  in  28  days  will  mow  3x4 
or  12  times  as  much  as  8  men  in  7  days. 

19.  How  many  bushels  of  wheat  will  serve  72  people 
8  days  when  4  bushels  serve  6  people  24  days  ? 

20.  If  2  horses  eat  8  bushels  of  oats  in  16  days,  how 
many  horses  will  eat  3000  bushels  in  24  days  ? 

21.  If  a  man  travels  150  miles  in  5  days  of  12  hours, 
in  how  many  days  of  10  hours  will  he  travel  500  miles  ? 

22.  If  939  soldiers  consume  351  bu.  of  wheat  in  21  days, 
how  many  soldiers  will  consume  1404  bu.  in  7  days  ? 

23.  If  5  men,  working  16  hours  a  day,  can  reap  a  field 
of  12£  acres  in  3-J-  days,  in  how  many  days  can  7  men, 
working  12  hours  a  day,  reap  a  field  of  15  acres  ? 

24.  If  7  men  in  8  days  of  11  hours  mow  22  acres,  in  how 
many  days  of  10  hours  will  12  men  mow  360  acres  ? 

25.  If  44  cannon,  firing  30  rounds  an  hour  for  3  hours 
a  day,  use  300  barrels  of  powder  in  5  days,  how  many  days 
will  400  barrels  last  66  cannon,  firing  40  rounds  an  hour 
for  5  hours  a  day  ? 


182  PROBLEMS. 


Areas. 

343.  If  the  length  and  breadth  of  a  rectangle  are 
expressed  in  the  same  linear  unit,  the  product  of  these 
two  numbers  will  express  its  area  in  square  units  of  the 
same  name  (§  145). 

Also,  the  number  of  square  units  in  a  rectangle  divided 
by  the  number  of  linear  units  in  one  dimension  gives  the 
number  of  linear  units  in  the  other  dimension  (§  145). 

Exercise  88. 

1.  Find  the  area  of  a  floor  16  ft.  3  in.  long  and  12  ft. 
6  in.  wide. 

Solution.     16  ft.  3  in.  =  16±  ft-;   12  ft.  6  in.  =  12*  ft. 
16*  X  12*  =  203*,  the  area  of  the  floor  in  square  feet. 

2.  A  rectangle  contains  672  sq.  ft.  108  sq.  in.,  and  is 
19  ft.  6  in.  wide.     Find  its  length. 

Solution.  672  sq.  ft.  108  sq.  in.  =  672f  sq.  ft. ;  19  ft.  6  in.  -  19*  ft. 
672f  -r  19*  =  34*,  the  length  of  the  rectangle  in  feet. 

3.  What  length  of  board  15  in.  wide  will  contain 
11  sq.  ft.  36  sq.  in.? 

4.  What  length  of  road  44  ft.  wide  will  contain  an  acre  ? 

5.  Find  the  area  of  a  rectangular  field  13.12  chains  long, 
10.35  chains  broad. 

6.  A  path  216  ft.  long  measures  72  sq.  yd.  Find  its 
breadth. 

7.  A  rectangular  field  of  21.66  A.  is  250.8  yd.  broad. 
Find  its  length. 

8.  What  is  the  area  of  a  table,  if  its  length  and  breadth 
are  4  ft.  3f  in.  and  2  ft.  9f  in.,  respectively  ? 

9.  From  each  corner  of  a  square,  each  side  of  which  is 
2  ft.  5  in.  long,  a  square  measuring  5  in.  on  a  side  is  cut 
out.     Find  the  area  of  the  remainder  of  the  figure. 


PROBLEMS.  183 

10.  The  length  and  breadth  of  a  map  are  4-J-  ft.  and 
3£  ft.,  respectively.  If  the  map  represents  77,760  sq.  mi. 
of  country,  how  many  square  miles  are  there  to  a  square 
inch? 

11.  In  rolling  a  grass  plot  that  is  24  yd.  long  and  con- 
tains 400  sq.  yd.,  how  many  times  must  a  roller  3  ft.  4  in. 
wide  be  drawn  over  it  lengthwise  that  the  whole  plot  may 
be  rolled  ? 

12.  How  many  sods,  each  2  ft.  3%  in.  long  and  8£  in. 
broad,  will  be  required  to  turf  an  acre  of  ground  ? 

13.  Find  the  area  of  a  picture  frame  2^  in.  broad,  if 
the  outside  measurement  is  4  ft.  6^  in.  in  length  and  2  ft. 
8  in.  in  width. 

14.  Find  the  expense  of  glazing  four  windows,  each  con- 
taining 12  panes,  if  the  panes  are  each  1  ft.  long  and 
10  in.  wide,  and  the  price  of  the  glass  is  38  cents  per 
square  foot. 

15.  A  field  76  yd.  long  and  56  yd.  broad,  enclosed  by  a 
wall,  has  a  border  4  ft.  wide  within  the  wall,  and  within 
this  a  path  5  ft.  wide.  If  the  remainder  of  the  field  is  grass, 
find  the  area  of  the  border,  of  the  path,  and  of  the  grass. 

16.  A  square  plot  of  land  127  yd.  long  has  a  path  1  yd. 
wide  running  round  the  inside  of  it.  Find  the  cost  of 
graveling  this  path  at  15  cents  per  square  yard. 

17.  A  street  f  of  a  mile  long  has  on  each  side  a  side- 
walk 7-J-  ft.  wide.  What  will  it  cost  to  pave  the  sidewalks 
with  stones,  each  measuring  2  ft.  9  in.  by  1  ft.  8  in.,  if  the 
stones  cost,  including  the  laying,  75  cents  each  ? 

18.  How  many  planks  11  ft.  by  9  in.  are  needed  to  cover 
a  platform  27  ft.  6  in.  long  and  8  yd.  wide  ?  What  will 
be  the  cost  at  20  cents  a  square  foot  ? 

19.  How  many  tiles  9  in.  long  and  4  in.  wide  will  be 
required  to  pave  a  walk  8  ft.  wide  that  surrounds  a  rect- 
angular court  60  ft.  long  and  36  ft,  wide  ? 


184  PROBLEMS. 

344.  If  the  diameter  of  a  circle  is  multiplied  by  3.1416, 
the  product  is  the  length  of  the  circumference  (§  138). 

If  the  circumference  of  a  circle  is  multiplied  by  0.31831, 
the  product  is  the  length  of  the  diameter  (§  139). 

345.  If  the  square  of  the  radius  of  a  circle  is  multiplied 
by  3.1416,  or  if  the  square  of  the  diameter  is  multiplied  by 
0.7854  (£  of  3.1416),  the  product  is  the  area  of  the  circle 
(§  146). 

20.  How  many  times  will  a  wheel  2£  ft.  in  diameter 
turn  in  going  a  distance  of  110  yards  ? 

21.  What  distance  will  a  wheel  ft  yd.  in  diameter  pass 
over  in  making  4£  revolutions  ? 

22.  Find  the  diameter  of  a  wheel  that  makes  9  revo- 
lutions in  going  7£  yards  ? 

23.  If  the  circumference  of  a  wheel  is  y  of  1  yd.  1|  ft., 
how  many  times  will  the  wheel  turn  in  going  3^  miles  ? 

24.  If  the  wheel  of  a  locomotive  is  3}  times  5.52  ft.  in 
circumference,  how  many  times  does  it  turn  in  a  minute 
when  the  locomotive  is  running  at  the  rate  of  13.34  mi. 
an  hour  ? 

25.  Find  the  area  of  a  circle  that  has  a  radius  of  3  feet. 

26.  What  is  the  area  of  a  circular  field  that  has  a  radius 
of  400  yards  ? 

27.  The  radius  of  the  rotunda  of  the  Pantheon  at  Rome 
is  71  ft.  6  in.     Find  the  area  of  the  floor. 

28.  The  diameter  of  a  cylindrical  cistern  is  13  ft.  What 
is  the  area  of  the  bottom  ? 

29.  The  two  dials  of  the  clock  of  St.  Paul's,  London, 
are  each  18}  ft.  in  diameter.  What  is  the  area  of  each  in 
square  feet  ? 

30.  At  20  cents  a  square  yard,  what  will  it  cost  to  gravel 
a  walk  6  ft.  wide  running  round  a  circular  fish  pond  70  yd. 
in  diameter  ? 


PROBLEMS.  185 

346.  If  the  square  of  the  diameter  of  a  sphere  is  multi- 
plied by  3.1416,  the  product  is  the  area  of  the  surface  of 
the  sphere  (§  149). 

31.  How  many  square  inches  on  the  surface  of  a  ball 
3  inches  in  diameter  ? 

32.  How  many  square  inches  of  surface  on  a  spherical 
blackboard  12  inches  in  diameter  ? 

33.  What  is  the  interior  surface  of  a  hemispherical  vase 
whose  interior  diameter  is  20  inches  ? 

34.  Find  the  external  and  the  internal  surface  of  a 
spherical  shell  whose  external  and  internal  diameters  are 
8  in.  and  5  in.,  respectively. 

35.  How  many  square  feet  of  tin  are  required  to  make 
16  hemispherical  bowls,  each  2  ft.  4  in.  in  diameter  ? 

347.  The  line  joining  the  centres  of  the  bases  of  a 
cylinder  is  called  the  axis  of  the  cylinder  (§  163). 

348.  Right  Cylinder.  A  cylinder  whose  axis  is  per- 
pendicular to  the  bases  is  called  a  right  cylinder. 

349.  To  Find  the  Area  of  the  Lateral  Surface  of  a 
Right  Cylinder, 

Multiply  the  height  of  the  cylinder  by  the  perimeter  of  its 
base. 

36.  Find  the  lateral  surface  of  a  right  cylinder  if  its 
height  is  10  in.  and  the  radius  of  its  base  is  7  in. 

37.  Find  the  lateral  surface  of  a  right  cylinder  if  its 
height  is  12  ft.  and  the  diameter  of  its  base  is  9  ft.  4  in. 

38.  At  32  cents  a  square  foot,  what  is  the  cost  of  cement- 
ing a  cylindrical  cistern  20  ft.  deep  and  18  ft.  in  diameter  ? 

39.  The  diameters  of  two  right  cylinders  of  the  same 
height  are  as  6  to  1.     Compare  the  lateral  surfaces. 


186  PROBLEMS. 

Carpeting  Rooms. 

350.  Carpeting  is  of  various  widths,  and  is  sold  by  the 
yard.     Oilcloth  and  linoleum  are  sold  by  the  square  yard. 

To  find  the  number  of  yards  of  carpeting  required  for 
a  room,  we  decide  whether  the  strips  shall  run  length- 
wise or  across  the  room,  and  then  find  the  number  of  strips 
required  (§  150).  The  number  of  yards  in  a  strip,  includ- 
ing the  waste  in  matching  the  pattern,  multiplied  by  the 
number  of  strips  will  give  the  number  of  yards  required. 

Exercise  89. 

1.  How  many  yards  of  carpeting  27  in.  wide  will  be 
required  for  a  floor  26  ft.  long,  15£  ft.  wide,  if  the  strips 
run  lengthwise  ?  How  many  if  the  strips  run  across  the 
room  ?     How  much  will  be  turned  under  in  each  case  ? 

2.  How  many  yards  of  carpeting  £■  yd.  wide  will  be 
required  for  a  room  8£  yd.  by  17  ft.,  if  the  strips  run 
lengthwise,  and  if  there  is  a  waste  of  ^  yd.  a  strip  ? 

3.  How  many  square  yards  of  oilcloth  will  be  required 
for  a  hall  floor  5£  yd.  long  and  10  ft.  wide  ? 

4.  At  $0.92  a  yard,  what  is  the  cost  of  a  carpet  27  in. 
wide  for  a  room  28 J  ft.  by  18f  ft.,  if  the  strips  run 
lengthwise  ? 

5.  Find  the  cost  of  carpet  30  in.  wide,  at  $1.25  per  yard, 
for  a  room  18  ft.  by  14  ft.,  if  the  strips  run  lengthwise. 

6.  Find  the  cost  of  carpeting  27  in.  wide,  at  $1.12£  per 
yard,  for  a  room  29  ft.  9  in.  by  23  ft.  6  in.,  if  the  strips 
run  across  the  room. 

7.  Find  the  cost  of  carpeting  27  in.  wide,  at  $2.75  per 
yard,  for  a  room  34  ft.  8  in.  by  13  ft.  3  in.,  if  the  strips 
run  lengthwise,  and  if  there  is  a  waste  of  £  yd.  a  strip. 

8.  Which  way  must  the  strips  of  carpet  27  in.  wide  run 
to  carpet  most  economically  a  room  20£  ft.  by  19£  ft.? 


PROBLEMS.  187 

Papering  Rooms. 

351.  Wall  paper  is  18  in.  wide,  and  is  sold  in  single 
rolls  8  yd.  long,  or  in  double  rolls  16  yd.  long. 

In  estimating  the  number  of  rolls  of  paper  required  for 
a  room  of  ordinary  height,  we  find  the  number  of  feet  in 
the  perimeter  of  the  room,  leaving  out  the  widths  of  the 
doors  and  windows,  and  allow  a  double  roll  or  two  single 
rolls  for  every  seven  feet. 

Note.  A  room  is  considered  of  ordinary  height  if  the  distance 
from  the  base  board  to  the  border  is  not  more  than  8  ft. 

9.  How  many  double  rolls  of  paper  will  be  required  for 
a  room  of  ordinary  height,  15  ft.  long  and  12  ft.  wide,  if 
the  room  has  one  door  and  three  windows,  each  3-J  ft.  wide  ? 

Solution.     Perimeter  of  room  =  2  X  (15  +  12)  ft.        =  54  f t. 

Width  of  door  and  windows  =  4  X  3±  ft.  =  14  ft. 

Perimeter  less  door  and  windows  =  40  ft. 

40 -S-  7  =  5^.     Hence,  6  double  rolls  will  be  required. 

10.  At  $2.25  a  double  roll,  put  on,  what  is  the  cost  of 
papering  a  room  of  ordinary  height,  16  ft.  by  14  ft.,  if  the 
room  has  two  doors  each  4  ft.  wide,  and  four  windows  each 

3  ft.  6  in.  wide  ? 

11.  At  75  cents  a  single  roll,  put  on,  what  is  the  cost  of 
papering  a  room  of  ordinary  height,  20  ft.  6  in.  long  and 
17  ft.  4  in.  wide,  if  the  room  has  two  doors  each  3  ft.  6  in. 
wide,  and  five  windows  each  3  ft.  3  in.  wide  ? 

12.  What  is  the  cost  of  the  border  for  the  room  of  Ex.  11 
at  $0.45  a  running  yard  ? 

Note.  Border  is  sold  by  the  yard,  and  no  allowance  is  made  for 
doors  or  windows. 

13.  At  $1.75  a  double  roll,  put  .on,  what  is  the  cost  of 
papering  a  room  of  ordinary  height,  18  ft.  6  in.  by  14  ft. 

4  in.,  if  the  room  has  three  doors  4  ft.  wide,  and  three 
windows  3  ft.  9  in.  wide  ? 


188  PROBLEMS. 

Plastering,  Painting,  and  Paving. 

352.  The  unit  of  plastering,  painting,  and  paving  is  the 
square  yard. 

The  rule  for  estimating  these  kinds  of  work  is  : 

Measure  the  total  area  ;  deduct  from  this  total  area  half 
the  area  of  the  doors,  windows,  and  other  openings,  and 
express  the  result  to  the  nearest  square  yard. 

14.  Find  at  20  cents  a  square  yard  the  cost  of  plaster- 
ing the  walls  and  ceiling  of  a  room  18  ft.  by  16  ft.  by  10  ft., 
if  the  room  has  two  doors  7  ft.  6  in.  by  4  ft.,  three  windows 
6  ft.  6  in.  by  4  ft.,  and  a  base  board  of  10  in. 

15.  Find  at  25  cents  a  square  yard  the  cost  of  plaster- 
ing the  walls  and  ceiling  of  a  room  16  ft.  by  15  ft.  by  10  ft., 
if  the  room  has  two  doors  7  ft.  by  3  ft.  9  in.,  three  windows 

5  ft.  6  in.  by  3  ft.  6  in.,  and  a  base  board  of  10  in. 

16.  Find  at  20  cents  a  square  yard  the  cost  of  plaster- 
ing the  walls  and  ceiling  of  a  room  15  ft.  by  14  ft.  by  9  ft. 

6  in.,  if  the  room  has  two  doors  7  ft.  4  in.  by  4  ft.,  two 
windows  5  ft.  6  in.  by  3  ft.  6  in.,  and  a  base  board  of 
9  in. 

17.  Find  at  15  cents  a  square  yard  the  cost  of  painting 
the  outside  of  the  walls  of  a  cottage-roofed  house  36  ft.  by 
32  ft.  by  13  ft.,  if  the  house  has  three  doors  7  ft.  6  in.  by 
4  ft.,  and  eleven  windows  6  ft.  by  4  ft. 

18.  Find  at  20  cents  a  square  yard  the  cost  of  painting 
the  walls  of  a  room  16  ft.  by  15  ft.  by  10  ft.,  if  the  room 
has  two  doors  7  ft.  6  in.  by  4  ft.,  four  windows  6  ft.  by 
3  ft.  9  in.,  and  a  base  board  of  9  in. 

19.  How  many  bricks  8  in.  long  and  4  in.  wide  will  be 
needed  to  pave  a  rectangular  court  60  ft.  by  30  ft.? 

20.  How  many  bricks  8  in.  long  and  2\  in.  thick,  laid 
on  edge,  will  be  needed  to  pave  the  court  of  Ex.  19  ? 


PROBLEMS.  189 

Clapboards  and  Shingles. 

353.  Clapboards.  Clapboards  are  usually  cut  4  ft.  long 
and  6  in.  wide,  and  laid  3^-  in.  to  the  weather.  Therefore, 
each  clapboard  covers  l£  sq.  ft.  of  surface. 

Note.  In  estimating  the  number  of  clapboards  required,  we 
deduct  the  area  of  all  openings. 

354.  Shingles.  Shingles  are  16  in.  long,  and  are  esti- 
mated to  average  4  in.  wide,  so  that  a  shingle  laid  4-J-  in.  to 
the  weather  will  cover  18  sq.  in.,  and  8  shingles  will  cover 
1  sq.  ft.  At  this  rate,  800  shingles  would  cover  a  square, 
or  100  sq.  ft.  It  is  found,  in  practice,  that  1000  shingles, 
laid  4-j-  in.  to  the  weather,  will  cover  about  120  sq.  ft. 

Shingles  are  put  up  in  bunches  of  250,  and  therefore  it 
takes  four  bunches  to  make  a  thousand. 

2 1 .  How  many  clapboards  will  be  required  for  the  front 
of  a  house  40  ft.  long  and  20  ft.  high,  allowing  120  sq.  ft. 
for  doors  and  windows  ? 

22.  How  many  clapboards  will  be  required  for  a  house 
44  ft.  long,  35  ft.  wide,  and  22  ft.  high  to  the  eaves,  if  the 
gables  extend  14  ft.  above  the  end  walls,  the  two  gables  to 
be  reckoned  as  one  full  wall,  and  500  sq.  ft.  to  be  allowed 
for  doors  and  windows  ? 

23.  Allowing  1000  shingles  for  120  sq.  ft.,  how  many 
thousand  will  be  required  for  the  pitched  roof  of  a  house 
60  ft.  long,  if  the  width  of  each  side  of  the  roof  is  24£  ft.? 

24.  Allowing  1000  shingles  for  110  sq.  ft.,  how  many 
thousand  will  be  required  for  the  pitched  roof  of  a  barn 
40  ft.  long,  if  the  width  of  each  side  of  the  roof  is  24  ft.? 

25.  Allowing  1000  shingles  for  120  sq.  ft.,  how  many 
thousand  will  be  required  for  the  pitched  roof  of  a  house 
28  ft.  long,  if  the  width  of  each  side  of  the  roof  is  18  ft.? 


190  PROBLEMS. 


Board  Measure. 

355.  Boards  one  inch  or  less  in  thickness  are  sold  by  the 
square  foot. 

Boards  more  than  one  inch  in  thickness,  and  all  squared 
lumber,  are  sold  by  the  number  of  square  feet  of  boards 
one  inch  in  thickness  to  which  they  are  equivalent. 

Thus,  a  board  16  ft.  long,  1  ft.  wide,  and  1  in.  thick  contains  16  ft. 
board  measure.  If  only  $,  f,  or  \  of  an  inch  thick,  it  still  con- 
tains 16  ft. ;  but  if  l±  in.  thick,  it  contains  1£  X  16,  or  20  ft.  board 
measure. 

356.  To  Find  the  Number  of  Feet  Board  Measure  in 
Boards  an  Inch  or  More  in  Thickness,  and  Squared  Lumber, 

Express  the  length  and  width  in  feet,  and  the  tlnckness 
in  inches  ;  take  the  product  of  these  three  numbers  for  the 
number  of  feet  board  measure. 

Note  1.  In  practice,  the  width  of  a  board,  unless  sawed  to  order, 
is  reckoned  only  to  the  next  smaller  half-inch.  Thus,  a  width  of 
llf  in.  is  reckoned  11  in.;  of  13|  or  13f  in.  is  reckoned  13$  in. 

Note  2.  If  a  board  tapers  regularly,  its  average  width  is  found  by 
taking  one  half  the  sum  of  its  end  widths. 

How  many  feet  board  measure  in : 

26.  A  board  18  ft.  long,  9  in.  wide,  £-  in.  thick  ? 

27.  A  board  16  ft.  long,  11  in.  wide,  1  in.  thick  ? 

28.  Twenty  boards  averaging  14  ft.  long,  10  in.  wide, 
1£  in.  thick  ? 

29.  Three  joists  13  ft.  long,  8  in.  wide,  3  in.  thick  ? 

30.  A  stick  of  timber  8  in.  by  9  in.,  and  27  ft.  long  ? 

31.  Two  beams,  each  6  in.  by  9  in.,  and  23  ft.  long? 

32.  Three  joists,  each  3  in.  by  4  in.,  and  11  ft.  long  ? 

33.  Five  joists,  each  6  in.  by  4  in.,  and  14  ft.  loug? 

34.  A  stick  of  timber  10  in.  square,  and  36  ft.  long  ? 

35.  Ten  planks,  each  13  ft.  long,  15  in.  wide,  2  in.  thick  ? 


PROBLEMS.  191 

Find  the  cost  of  : 

36.  Nine  joists,  each  15  ft.  long,  3£  in.  by  5  in.,  at  $12 
per  M. 

Note.     The  abbreviation  per  M  means  by  the  thousand. 

37.  Thirty  planks,  each  12  ft.  long,  11  in.  wide,  3  in. 
thick,  at  $15  per  M. 

38.  Four  sticks  of  timber,  each  8  in.  by  9  in.  and  23  ft. 
long,  at  $18  per  M. 

39.  A  board  24  ft.  long,  23  in.  wide  at  one  end  and 
17  in.  at  the  other,  and  1£  in.  thick,  at  $30  per  M. 

40.  A  stick  of  timber  29  ft.  long,  10  in.  by  12  in.,  at 
$13.50  per  M. 

41.  The  flooring  for  two  floors,  each  23  ft.  by  17  ft., 
each  floor  double,  and  of  boards  j-  in.  thick;  the  under 
floors  at  $18,  and  the  upper  at  $24,  per  M. 

42.  The  flooring  timbers  for  a  room  23  ft.  by  17  ft.,  at 
$18  per  M,  if  they  are  2  in.  by  10  in.,  17  ft.  long,  and  are 
placed,  on  edge,  one  close  to  each  wall  and  the  others  with 
spaces  ||  ft.  wide  between  them. 

357.  Round  Logs.  Round  logs  are  sold  by  the  number 
of  feet  board  measure  that  can  be  cut  from  them.  If  a  log 
is  not  more  than  16  ft.  long,  we  measure  the  length  of  the 
log  and  the  diameter  of  the  smaller  end,  and  find  the  number 
of  feet  board  measure  as  follows  : 

Subtract  twice  the  diameter  expressed  in  inches  from  the 
square  of  the  diameter,  and  take  f  i  of  the  remainder  for  the 
number  of  feet  board  measure  in  a  log  10  ft.  long. 

43.  Find  the  number  of  feet  board  measure  in  a  log  12  ft. 
long,  and  20  in.  in  diameter  at  the  smaller  end. 

Solution.  202  -  2  X  20  =  400  -  40  =  360. 

f  J-  of  360  =  189. 
As  the  log  is  12  ft.  long,  we  must  take  f$  of  189  ft.  to  obtain  the 
number  of  feet  in  the  whole  log ;  that  is,  226.8  ft. 


192  PROBLEMS. 

By  this  rule  find  the  number  of  feet  board  measure  in  : 

44.  A  log  14  ft.  long,  smallest  diameter  17  in. 

45.  A  log  11  ft.  long,  smallest  diameter  13  in. 

46.  A  log  16  ft.  long,  smallest  diameter  20  in. 

47.  A  log  12  ft.  long,  smallest  diameter  15  in. 

Find  the  value  at  $9  per  M  of  : 

48.  A  log  15  ft.  long,  smallest  diameter  11  in. 

49.  A  log  16  ft.  long,  smallest  diameter  13  in. 

50.  A  log  13  ft.  long,  smallest  diameter  16  in. 

51.  A  log  14  ft.  long,  smallest  diameter  12  in. 

358.  Large,  heavy  timber  of  hard  wood  is  generally 
sold  by  the  ton,  signifying  50  cu.  ft.,  or  600  ft.  board 
measure. 

Volumes. 

359.  If  the  length,  breadth,  and  height  of  a  rectangular 
solid  are  expressed  in  the  same  linear  unit,  the  product  of 
these  three  numbers  will  express  its  volume  in  cubic  units 
of  the  same  name  (§  161). 

Also,  the  number  of  cubic  units  in  a  rectangular  solid 
divided  by  the  product  of  the  numbers  of  linear  units  in 
any  two  dimensions  gives  the  number  of  linear  units  in  the 
third  dimension  (§  161). 

Exercise  90. 

1.  Find  the  volume  of  a  rectangular  solid  7  ft.  long, 
2  ft.  6  in.  wide,  and  11  in.  thick. 

2.  How  many  cubic  feet  of  air  in  a  hall  54  ft.  long, 
33  ft.  wide,  and  21  ft.  4  in.  high  ? 

3.  Find  the  volume  of  a  cube  whose  edge  is  2-j-  yd. 

4.  A  cellar  was  dug  21  ft.  long,  17  ft.  3  in.  wide,  and 
9  ft.  deep.  How  many  cubic  yards  of  earth  were  taken 
out? 


PROBLEMS.  193 

5.  Find  the  volume  of  a  brick  8  in.  long,  3£  in.  wide, 
and  (1\  in.  thick. 

6.  How  many  cubic  feet  of  water  will  a  rectangular 
cistern  hold  whose  length,  breadth,  and  height  are  5  ft. 
4  in.,  3  ft.  6  in.,  and  2  ft.  10  in.,  respectively  ? 

7.  Find  the  volume  in  cubic  inches  of  a  bar  of  iron 
21  ft.  long,  3  in.  wide,  and  2  in.  thick. 

8.  What  is  the  value  at  $190  a  cubic  inch  of  a  bar  of 
gold  8  in.  long  and  §  of  an  inch  square  ? 

9.  A  rectangular  reservoir  15  yd.  long,  12  yd.  wide, 
holds  330  cu.  yd.  of  water.     What  is  its  depth  ? 

10.  What  length  must  be  cut  off  a  beam  9  in.  by  15  in. 
that  the  part  cut  off  may  contain  2£  cu.  ft.? 

11.  How  high  is  a  room,  if  it  is  31  ft.  3  in.  long,  24  ft. 
broadband  contains  10,000  cu.  ft.  of  air? 

12.  A  piece  of  wood  5  ft.  long,  1  ft.  broad,  and  9  in. 
thick  is  cut  up  into  matches  2-J-  in.  long  and  0.1  of  an  inch 
square.  How  many  matches  will  there  be,  if  no  allowance 
is  made  for  waste  in  cutting  ? 

13.  How  long  a  wall  6  ft.  high,  12f  in.  thick,  can  be 
built  with  the  bricks  forming  a  rectangular  pile  17  ft.  6  in. 
long,  5  ft.  wide,  and  4  ft.  3  in.  high  ? 

14.  Find  the  surface  of  a  cube  whose  edge  is  3  ft.  5f  in. 

15.  Find  the  surface  of  a  rectangular  block  of  stone 
4  ft.  long,  2\  ft.  broad,  and  \\  ft.  thick. 

16.  A  lake  whose  area  is  45  A.  is  covered  with  ice  3  in. 
thick.  Find  the  weight  of  the  ice  in  tons,  if  a  cubic  foot 
of  ice  weighs  920  oz. 

17.  How  many  bricks  will  be  required  to  build  a  wall 
75  ft.  long,  6  ft.  high,  and  16  in.  thick,  if  each  brick  is 
8  in.  long,  4  in.  wide,  and  2\  in.  thick  ? 

18.  The  ceiling  of  a  room  27  ft.  long,  24  ft.  broad,  and 
10  ft.  high  is  to  be  raised  so  as  to  increase  the  space  by 
84  cu.  yd.     What  will  then  be  the  height  of  the  room  ? 


194  PROBLEMS. 

19.  Find  the  cost  of  making  a  road  110  yd.  long  and 
18  ft.  wide,  if  the  soil  is  first  removed  to  the  depth  of  1  ft. 
at  a  cost  of  25  cents  a  cubic  yard,  rubble  then  laid  8  in. 
deep  at  25  cents  a  cubic  yard,  and  gravel  placed  on  top 
9  in.  thick  at  62 J  cents  a  cubic  yard. 

20.  If  a  rectangular  block  of  wood  5  ft.  4.8  in.  long, 
1  ft.  9  in.  wide  and  thick,  weighs  7.56  cwt.,  find  in  pounds 
its  weight  per  cubic  foot. 

360.  A  cord  of  wood  or  stone  is  a  pile  8  ft.  long,  4  ft. 
wide,  and  4  ft.  high,  making  128  cu.  ft. 

A  cord  foot  is  a  pile  1  ft.  long,  4  ft.  wide,  and  4  ft.  high, 
and  is  therefore  one  eighth  of  a  cord,  or  16  cu.  ft.      Hence, 


Cord.  Cord  Foot. 

361.   To  Find  the  Number  of  Cords  in  a  Pile  of  Wood, 

Divide  the  product  of  the  length,   breadth,   and  height, 
expressed  in  feet,  by  8  X  ^  X  4- 

21.  How  many  cords  of  wood  in  a  pile  40  ft.  long,  4  ft. 
wide,  and  5  ft.  4  in.  high  ? 

22.  A  pile  of  wood  containing  67£  cords  is  270  ft.  long 
and  4  ft.  wide.     How  high  is  it  ? 

23.  What  will  be  the  cost  of  a  pile  of  wood  25  ft.  long, 
4  ft.  wide,  and  4  ft.  8  in.  high,  at  $3.75  a  cord  ? 


PROBLEMS.  195 

24.  What  must  be  the  length  of  a  load  of  wood  3£  ft. 
high  and  5  ft.  wide  to  contain  a  cord  ? 

25.  How  high  must  manure  be  piled  in  a  cart  6  ft.  by 
4  ft.,  that  the. load  may  contain  half  a  cord  ? 

26.  How  many  cords  of  wood  in  a  pile  32  ft.  long,  8  ft. 
wide,  and  6  ft.  high  ? 

27.  How  many  cords  of  wood  in  a  pile  40  ft.  long,  4  ft. 
wide,  and  8  ft.  high  ? 

28.  Find  the  cost  of  the  wood  at  $3.75  a  cord  that  can 
be  piled  in  a  shed  18  ft.  long,  16  ft.  wide,  and  7  ft.  high. 

362.  By  §  162,  to  find  the  volume  of  a  sphere,  we  mul- 
tiply the  cube  of  the  diameter  by  0.5236  (J  of  3.1416). 

29.  Find  the  number  of  cubic  inches  in  a  sphere  11  in. 
in  diameter. 

30.  How  many  cubic  inches  of  water  can  be  poured  into 
a  hollow  sphere  whose  inner  diameter  is  16J  in.? 

31.  What  is  the  volume  of  the  ball  on  top  of  St.  Paul's 
in  London,  which  is  6  ft.  in  diameter  ? 

32.  If  30  cu.  in.  of  powder  weigh  1  lb.,  how  many 
ounces  of  powder  will  just  fill  a  shell,  inner  diameter  3  in.? 

363.  To  find  the  volume  of  a  cylinder,  we  multiply  the 
number  of  square  units  in  its  base  by  the  number  of  linear 
units  of  the  same  name  in  its  height  (§  164). 

33.  Find  the  volume  of  a  cylinder  whose  height  is  5  ft. 
and  the  radius  of  whose  base  is  1  ft.  2  in. 

34.  Find  the  volume  of  a  cylinder  whose  height  is  4  ft. 
6  in.  and  the  diameter  of  whose  base  is  8  ft.  2  in. 

35.  How  many  cubic  yards  of  earth  must  be  excavated 
to  make  a  well  3  ft.  in  diameter  and  20  ft.  deep  ? 

36.  How  many  cubic  yards  in  a  tunnel  800  ft.  long,  if  a 
cross  section  is  a  semicircle  with  a  radius  of  10  ft.? 


196  PROBLEMS. 

Capacity  of  Bins  and  Cisterns. 

37.  Find  the  number  of  cubic  feet  in  a  bushel. 

Solution.     Since  a  bushel  contains  2160.42  cu.  in.  (§  294),  and  a 

2160  42 
cubic  foot  contains  1728  cu.  in.,  therefore,  a  bushel  contains    ,    ' 

1728 

cu.  ft.,  or  1.24445  cu.  ft. 

If  we  add  i  of  0.01  of  1.24446  to  1.24445,  we  have  1.25+ .     Hence, 

364.  To  Find  the  Approximate  Number  of  Bushels  a 
Bin  will  Hold, 

Take  $  of  the  number  of  cubic  feet  in  the  bin,  and  add  to 
the  product  £  of  0.01  of  the  product. 

365.  To  Find  the  Number  of  Cubic  Feet  Required  for 
a  Given  Number  of  Bushels, 

Take  |  of  the  number  of  bushels,  and  subtract  from  the 
product  ^  of  0.01  of  the  product. 

38.  Find  the  number  of  bushels  a  bin  will  hold  that  is 
6  ft.  long,  5  ft.  wide,  and  4  ft.  deep. 

Solution.  f  of  6x5x4  =  96. 

£  of  0.01  of  96  s    0.48 
96.48 
Therefore,  the  bin  will  hold  96.48  bu. 

39.  Find  the  number  of  cubic  feet  required  for  1000  bu. 

Solution.  £  of  1000  =  1250. 

£  of  0.01  of  1250  =       6.25 
1243.75 
Hence,  1243.75  cu.  ft.  are  required  for  1000  bu. 

40.  Find  the  number  of  bushels  a  bin  will  hold  that  is 

8  ft.  long,  4  ft.  wide,  3  ft.  deep. 

41.  Find  the  number  of  bushels  a  bin  will  hold  that  is 

9  ft.  long,  6  ft.  6  in.  wide,  3  ft.  4  in.  deep. 

42.  Find  the  depth  of  a  bin  that  will  hold  360  bu.,  if  its 
length  is  12  ft.  and  its  width  6  ft. 


PROBLEMS.  197 

43.  Find  the  length  of  a  bin  that  is  6  ft.  wide  and  5  ft. 
deep,  if  it  will  hold  400  bu. 

44.  Find  the  number  of  bushels  that  will  fill  a  bin  8.5  ft. 
long,  4.5  ft.  wide,  3.5  ft.  deep. 

45.  A  bin  20  ft.  long,  12  ft.  wide,  and  6  ft.  deep  is  full 
of  wheat.     What  is  its  value  at  $0.75  a  bushel  ? 

46.  If  a  ton  of  coal  occupies  40  cu.  ft.,  how  many  tons 
of  coal  will  fill  a  bin  21  ft.  long,  10  ft.  wide,  5  ft.  deep  ? 

47.  If  a  ton  of  Lehigh  coal  occupies  35  cu.  ft.,  how 
many  tons  of  Lehigh  coal  will  fill  a  bin  8  ft.  long,  5  ft. 
9  in.  wide,  3  ft.  6  in.  deep  ? 

48.  How  many  bushels  will  a  bin  hold  that  is  22  ft.  long, 
12  ft.  6  in.  wide,  9  ft.  9  in.  deep  ? 

49.  Find  the  number  of  gallons  in  a  cubic  foot. 

Solution.  Since  a  cubic  foot  contains  1728  cu.  in.,  and  a  gallon 
contains  231  cu.  in.,  therefore,  a  cubic  foot  contains  -Vit5"  S^m  or 
7.480&2  gal. 

If  we  add  J  of  0.01  of  7.48052  to  7.48052,  we  have  7.5  nearly. 
Hence, 

366.  To  Find  the  Approximate  Number  of  Gallons  a 
Cistern  will  Hold, 

Multiply  the  number  of  cubic  feet  by  7J,  and  from  the 
product  subtract  \  of  0.01  of  the  product. 

367.  To  Find  the  Exact  Number  of  Gallons  a  Cistern 
will  Hold, 

Divide  the  number  of  cubic  inches  in  the  contents  of  the 
cistern  by  231. 

50.  Find  the  exact  number  of  gallons  a  cistern  will  hold 
that  is  5  ft.  square,  and  6  ft.  deep. 

51.  Find  the  exact  number  of  gallons  a  cistern  will  hold 
that  is  13  ft.  long,  6  ft.  wide,  7  ft.  4  in.  deep. 

52.  Find  the  exact  number  of  gallons  a  tank  will  hold 
that  is  4  ft.  long,  2  ft.  8  in.  wide,  1  ft.  8  in.  deep. 


198  PROBLEMS. 

53.  Find  the  capacity  in  cubic  feet  of  a  cistern  that  will 
hold  200  bbl.  of  water. 

54.  Find  the  approximate  number  of  gallons  a  cylindri- 
cal cistern  will  hold  that  is  6  ft.  in  diameter  and  7  ft.  deep. 

55.  Find  the  approximate  number  of  gallons  a  cylindri- 
cal vessel  will  hold  that  is  12  in.  in  diameter  and  10  in.  deep. 

56.  How  many  quarts  will  a  cylindrical  vessel  hold 
5£  in.  in  diameter  and  6  in.  deep  ? 

57.  How  many  quarts  will  a  hollow  sphere  hold  whose 
interior  diameter  is  12  in.? 

58.  What  part  of  a  bushel  will  a  hemispherical  bowl 
hold  that  is  13  in.  in  diameter  ? 

59.  If  a  cubical  box  2  ft.  on  an  edge  contains  a  solid 
sphere  2  ft.  in  diameter,  how  many  gallons  of  water  can  be 
poured  into  the  box  ? 

60.  If  64  qt.  of  water  are  poured  into  a  vessel  that  will 
hold  2  bu.  of  wheat,  what  part  of  the  vessel  will  be  filled  ? 

Specific  Gravity. 

368.  The  specific  gravity  of  a  substance  is  the  number 
found  by  dividing  the  weight  of  the  substance  by  the 
weight  of  an  equal  bulk  of  water  (§  165). 

Note.  If  a  substance  is  in  water,  the  water  buoys  it  up  by  just 
the  weight  of  the  water  displaced  by  it. 

Exercise  91. 

1 .  Find  the  number  of  cubic  inches  in  1  oz.  (av.)  of  water. 

2.  Find  the  weight  in  ounces  (av.)  of  1  on.  in.  of  water. 

3.  Find  the  weight  in  ounces  (av.)  of  1  pt.  of  water. 

4.  Find  the  number  of  pints  in  1  lb.  of  water. 

5.  Find  the  weight  in  grains  of  1  cu.  in.  of  water. 

6.  A  bar  of  iron  5  in.  long  and  2  in.  square  weighs  5  lb. 
What  is  the  specific  gravity  of  the  iron  ? 


PROBLEMS.  199 

7.  If  a  bar  of  iron  18  in.  long,  2-J-  in.  wide,  If  in.  thick, 
weighs  18  lb.  9  oz.,  what  is  the  specific  gravity  of  the  iron  ? 

8.  If  the  specific  gravity  of  iron  is  7.48,  find  the  num- 
ber of  cubic  inches  of  iron  to  the  pound. 

9.  If  the  specific  gravity  of  gold  is  19.36,  find  the  num- 
ber of  cubic  inches  in  2  lb.  6£  oz.  of  gold. 

10.  How  many  pounds  does  a  boy  lift  in  raising  a  cubic 
foot  of  stone  under  water,  if  its  specific  gravity  is  2£  ? 

11.  A  square-built  scow  12  ft.  long,  6  J  ft.  wide,  sinks 

5  in.  into  the  water.     What  does  it  weigh,  and  how  many 
pounds  will  be  required  to  sink  it  7  in.  deeper  ? 

12.  A  square-built  scow  11  ft.  long,  5^  ft.  wide,  weighs 
320  lb.,  and  is  loaded  with  750  lb.  of  stone.  How  deep 
does  it  sink  in  the  water  ? 

13.  How  many  tons  of  ice,  specific  gravity  0.93,  can  be 
packed  in  a  building  50  ft.  long,  40  ft.  wide,  20  ft.  high  ? 

14».  If  the  specific  gravity  of  an  iceberg  is  0.9,  how  many 
cubic  yards  does  an  iceberg  contain  that  is  40  rd.  long, 

6  yd.  wide,  and  rises  160  ft.  out  of  the  sea  ? 

15.  If  a  cubic  foot  of  brick  wall  weighs  90  lb.  and  con- 
tains 22  bricks,  with  the  mortar,  what  is  the  weight  and 
the  specific  gravity  of  a  brick  and  its  share  of  mortar  ? 

16.  What  is  the  weight  of  a  brick  wall  40  ft.  long,  20  ft. 
high,  and  1  ft.  thick,  if  the  specific  gravity  of  a  brick  with 
its  mortar  is  1.46  ?  How  many  thousand  bricks  will  be 
required  for  the  wall,  allowing  22  for  a  cubic  foot  ? 

17.  If  the  specific  gravity  of  iron  is  7.48,  what  is  the 
weight  of  a  cylindrical  iron  shell  1  in.  thick  and  2  ft.  long, 
whose  inner  radius  is  7  in.? 

18.  If  a  piece  of  marble  weighs  37.78  oz.  in  air,  and 
23.89  oz.  in  water,  what  is  its  volume  and  its  specific 
gravity  ? 

19.  If  a  mass  of  lead  weighs  1986^  lb.  in  air,  and  1811 J  lb. 
in  water,  what  is  its  volume  and  its  specific  gravity  ? 


200 


PROBLEMS. 


210— 

1~^1 

200— 

190— 

—90 

180— 

—80 

170— 

160— 

—70 

150— 

140— 

—60 

130— 

120— 

—50 

110 — 

100 — 

—40 

90 — 

—30 

80— 

70— 

—20 

60— 

50— 

—10 

40— 

30— 

—0 

20— 

—10 

10— 

0— 

—20 

10— 

20— 

—30 

30— j 

Temperature. 

369.  A  thermometer  is  an  instrument 
for  registering  temperature. 

There  are  three  scales  for  registering  tem- 
perature by  means  of  the  thermometer. 

Fahrenheit's  has  the  freezing  point  of 
water  marked  32°,  and  boiling  point  212°. 

The  Centigrade  lias  the  freezing  point  0°, 
and  the  boiling  point  100°. 

Reaumur's  has  the  freezing  point  0°,  and 
the  boiling  point  80°. 

Temperature  below  0°  is  indicated  by  pre- 
fixing the  minus  sign. 

Thus,  —  20°  means  20°  below  zero. 


370.   Examples. 

Reaumur's  scale. 


1.   Express  80°  C.  in 


Therefore, 


100°  C.  =  80°  R. 


=  64°  R. 


Thermometer. 


2.  Express  50°  F.  in  Centigrade  scale. 

The  number  of  degrees  F.  between  the  freezing 
and  boiling  points  is  180. 

Therefore,  1°  F.  =  j§°°  C.  =  $°  C. 

But  50°  F.  is  18°  F.  above  freezing  point, 
and  $  of  18°  =  10°. 

That  is,     50°  F.  =  10°  C. 

3.  Express  60°  C.  in  Fahrenheit's  scale. 

100°  C.  =  180°  F. 
Therefore,      60°  C.  =  ^  of  180°  F. 

=  108°  F. 
This  is  the  height  above  the  freezing  point,  and 
is  marked  32°  +  108°  =  140°. 


PROBLEMS.  201 

Exercise  92. 
Express : 

1.  59°  F.  in  Centigrade  scale  ;    in  Reaumur's  scale. 

2.  77°  F.  in  Centigrade  scale  j    in  Reaumur's  scale. 

3.  950°  F.  in  Centigrade  scale ;    in  Reaumur's  scale. 

4.  —  40°  F.  in  Centigrade  scale  ;    in  Reaumur's  scale. 

5.  —  4°  F.  in  Centigrade  scale  ;    in  Reaumur's  scale. 

6.  10°  C.  in  Fahrenheit's  scale  ;    in  Reaumur's  scale. 

7.  22°  C.  in  Fahrenheit's  scale  ;    in  Reaumur's  scale. 

8.  — 30°  C.  in  Fahrenheit's  scale  ;    in  Reaumur's  scale. 

9.  —  llf  °  C.  in  Fahrenheit's  scale  ;  in  Reaumur's  scale. 

Time  and  Work  Problems. 
Exercise  93. 

1,  If  one  man  can  do  a  piece  of  work  in  9  days  and 
another  man  can  do  the  same  work  in  8  days,  in  how  many 
days  can  the  men  working  together  do  the  work  ? 

Solution.  If  a  man  can  do  a  piece  of  work  in  9  days,  in  1  day  he 
can  do  £  of  the  work ;  and  if  another  man  can  do  the  same  work  in 
8  days,  in  1  day  he  can  do  f  of  it. 

Both  men  together  in  1  day  can  do  ^  +  },  or  \\  of  the  work. 

Therefore,  if  the  whole  work  is  considered  as  divided  into  72  equal 
parts,  they  together  can  do  17  of  these  parts  in  1  day,  and  the  number 
of  days  required  to  do  the  whole  work  will  be  $f,  or  4T4r. 

2.  A  cistern  can  be  filled  by  a  water-pipe  in  30  min. 
and  emptied  by  a  waste-pipe  in  20  min.  If  the  cistern  is 
full  and  both  pipes  are  opened,  in  how  many  minutes  will 
the  cistern  be  emptied  ? 

Solution.     In  1  min.  the  waste-pipe  empties  fa  of  the  cistern. 
In  1  min.  the  water-pipe  fills  fa  of  the  cistern. 
When  both  are  open  fa  —  fa,  or  fa  of  the  cistern,  is  emptied  in 
1  min. 

Therefore,  the  cistern  will  be  emptied  in  60  min. 


202  PROBLEMS. 

3.  If  A  can  mow  a  certain  meadow  in  4  days,  and  B  in 

3  days,  how  long  will  it  take  both  together  ? 

4.  If  A  can  lay  a  certain  wall  in  4£  days,  and  B  in  5 J 
days,  how  long  will  it  take  both  together  ? 

5.  If  one  pipe  will  fill  a  cistern  in  4 J  hr.,  and  another 
pipe  in  3£  hr.,  how  long  will  it  take  both  together  to  fill 
the  cistern  ? 

6.  If  A  can  go  from  Boston  to  Albany  in  9 J  hr.,  and  B 
from  Albany  to  Boston  in  11^-  hr.,  and  they  start  at  the 
same  time,  in  how  many  hours  will  they  meet  ? 

7.  If  it  takes  A  working  alone  4  days,  B  3  days,  and 
C  4£  days  to  do  a  piece  of  work,  how  long  will  it  take  to 
do  the  work  if  all  three  work  together  ? 

8.  A  can  mow  |  of  a  field  in  3  days  ;   and  B  £  of  it  in 

4  days.     How  long  will  it  take  both  together  to  mow  the 
field? 

9.  One  pipe  can  fill  a  cistern  half  full  in  £  of  an  hour, 
and  another  can  fill  it  three  quarters  full  in  £  an  hour. 
How  long  will  it  take  both  pipes  together  to  fill  the  cis- 
tern? 

10.  A  pipe  can  fill  a  cistern  one  third  full  in  £  of  an 
hour  ;  a  waste-pipe  can  empty  one  fourth  of  the  cistern  in 
20  minutes.  If  both  pipes  are  opened,  in  what  time  will 
the  cistern  be  filled  ? 

11.  A  cistern  that  will  hold  100  gallons  can  be  filled 
by  a  pipe  in  25  minutes  and  emptied  by  a  waste-pipe  in 
45  minutes.  If  the  cistern  is  empty  and  both  pipes  are 
opened,  how  long  will  it  take  to  fill  the  cistern,  and  how 
much  water  will  be  wasted  ? 

12.  If  water  runs  into  a  cistern  by  one  pipe  at  the  rate 
of  2  gal.  in  3  min.,  by  another  at  the  rate  of  5  gal.  in 

4  min.,  and  runs  out  by  a  third  at  the  rate  of  4  gal.  in 

5  min.,  how  long  will  it  take  to  gain  71  gal.  in  the  cis- 
tern ? 


PROBLEMS.  203 

13.  A  can  do  a  piece  of  work  in  6  days,  and  B  can  do  it 
in  7  days.  If  they  work  together  2  days,  and  A  then 
leaves,  how  long  will  it  take  B  to  finish  the  work  ? 

14.  A  cistern  that  will  hold  200  gal.  has  two  pipes  ;  one 
will  supply  0.15  gal.  a  second,  the  other  If  qt.  a  second. 
If  the  first  is  turned  on  for  10  minutes  and  afterwards  both 
run  together,  in  what  time  will  the  cistern  be  filled  ? 

15.  A  and  B  together  can  do  a  piece  of  work  in  15  days. 
After  working  together  6  days,  A  leaves  and  B  finishes  the 
work  in  30  days  more.  In  how  many  days  can  each  alone 
do  the  work  ? 

16.  A  and  B  together  can  do  a  piece  of  work  in  12  days. 
After  working  together  9  days,  however,  they  call  in  C  to 
help  them,  and  the  three  finish  the  work  in  2  days.  In 
how  many  days  can  C  alone  do  the  work  ? 

17.  A  and  B  together  can  do  a  piece  of  work  in  2\  days  ; 
A  and  C  in  3-J-  days  ;  B  and  C  in  3f  days.  How  long  will 
it  take  the  three  working  together  to  do  the  work,  and  how 
long  will  it  take  each  alone  ? 

Note.     By  working  two  days  each  they  can  do  —  +  ^r  +  7^  of  the 

2f      o-j-      of- 

work,  that  is,  f  +  T%  +  T\  or  ||  of  the  work.     Hence,  by  working 

one  day  each,  they  can  do  \  of  f  § ,  or  f  $  of  the  work. 

In  one  day  A  can  do  §£  —  T\  of  the  work. 

18.  A  and  B  together  can  do  a  piece  of  work  in  48  days  ; 
lA  and  C  together  in  30  days  ;   B  and  C  together  in  26f 

days.     How  long  will  it  take  each  alone  to  do  the  work  ? 

19.  A  cistern  has  three  pipes.  The  first  and  second  will 
fill  it  in  1  hr.  10  min. ;  the  first  and  third  in  1  hr.  24  min. ; 
the  second  and  third  in  2  hr.  20  min.  How  long  will  it 
take  each  alone  to  fill  the  cistern  ? 

20.  A,  B,  and  C  together  can  do  a  piece  of  work  in  10 
days  ;  A  and  B  together  in  12  days  ;  B  and  C  together  in 
20  days.     How  long  will  it  take  each  alone  to  do  the  work  ? 


204  PROBLEMS. 

Rate  and  Time  Problems. 
Exercise  94. 

1.  A  train  travels  24  miles  in  0.8  of  an  hour.  Find 
its  rate  per  hour. 

Solution.  If  the  question  had  read,  a  train  travels  70  mi.  in  2  hr., 
its  rate  per  hour  would  be  found  by  dividing  the  whole  distance, 
70  mi.,  by  2.  The  application  of  the  same  method  to  this  question 
gives  24  mi.  -f  0.8,  or  30  mi.,  for  the  rate  per  hour. 

2.  A  train  runs  from  New  York  to  Philadelphia,  90 
miles,  in  1  hr.  33  min.     What  is  its  rate  per  hour  ? 

Solution.  1  hr.  33  min.  =  1££  hr.  Therefore,  the  rate  per  hour 
is  90  mi.  -f  1£$,  or  58 &  mi.     Hence, 

371.  To  Find  the  Rate  when  the  Distance  and  the 
Time  are  Known, 

Divide  the  distance  by  the  number  of  units  of  time. 

3.  A  train  runs  from  New  York  to  Philadelphia,  90 
miles,  in  2  hr.  5  min.     What  is  its  rate  per  hour  ? 

4.  Winlock,  in  1869,  found  that  electricity  went 
through  7200  miles  of  wire  in  §-  of  a  second.  What 
was  its  rate  per  second  ? 

5.  If  the  time  required  for  a  signal  to  pass  through 
the  cable  from  Brest  to  Duxbury,  3799  miles,  is  0.816  of 
a  second,  what  is  the  rate  per  second  ? 

6.  If  the  report  of  a  gun  1\  miles  distant  is  heard 
in  5f  seconds  after  the  flash  is  seen,  what  is  the  velocity  of 
sound  in  feet  per  second  ? 

7.  If  a  man  walks  3 J  miles  in  46  minutes,  what  is 
his  rate  per  hour  ? 

8.  If  a  horse  goes  48  miles  in  10  hr.  40  min.,  what  is 
his  average  rate  per  hour  ? 

9.  If  a  stone  on  a  glacier  is  carried  95 J  feet  in  188 
days,  what  is  its  rate  in  inches  per  day  ? 


PROBLEMS.  205 

10.  If  a  horse  went  5£  miles  in  33  minutes,  how  long 
did  it  take  him  to  go  a  mile  ? 

Solution.     33  min.  -r  5£  =  6  min.     Hence, 

372.  To  Find  the  Rate  for  a  Unit  of  Distance  when 
the  Distance  and  the  Time  are  Known, 

Divide  the  time  by  the  number  of  units  of  distance. 

11.  If  a  horse  can  trot  |  of  a  mile  in  2 J  minutes,  in 
what  time  can  he  trot  a  mile  ? 

12.  If  a  train  runs  18  miles  in  39  minutes,  how  long 
does  it  take  to  run  one  mile  ? 

13.  If  sound  travels  1125  feet  a  second,  how  long  will  it 
take  to  travel  one  mile  ? 

14.  If  a  train  requires  3  hours  to  run  104£  miles,  find 
its  average  time  for  running  a  mile. 

15.  If  a  man  cuts  1\  A.  of  grass  in  3  J-  days,  what  part 
of  a  day  will  it  take  him  to  cut  an  acre  ?  If  10  hr.  makes 
a  day,  what  part  of  an  acre  will  he  cut  in  an  hour  ? 

16.  If  a  mower  cuts  3^  square  rods  in  -J  of  an  hour,  how 
many  acres  will  he  cut  in  a  day  of  10  hours  ? 

17.  If  a  fountain  yields  117^-  gallons  of  Water  in  f  of 
an  hour,  at  what  rate  per  hour  is  the  water  flowing  ? 

18.  If  a  merchant's  profits  are  $3147  in  1\  months, 
what  will  be  his  profits  at  the  same  rate  for  a  year  ? 

19.  If  a  wheel  turns  17°  30'  in  35  minutes,  in  how  many 
hours  does  it  make  a  complete  revolution  ? 

20.  If  a  man's  expenditures  are  $4358  in  13-J-  months, 
what  is  his  yearly  rate  of  expenditure  ? 

21.  If  a  cistern  loses  by  leakage  7  gal.  1  pt.  in  49  hr. 
40  min.,  what  is  its  hourly  rate  of  loss  ? 

22.  If  a  man  travels  3f  miles  in  7£  minutes,  how  many 
miles  will  he  travel  in  50  minutes  ?  How  long  will  it  take 
him  to  travel  50  miles  ? 


206  PROBLEMS. 

Clock  Problems. 
Exercise  95. 

1 .  At  what  time  between  5  and  6  o'clock  do  the  hour 
and  minute  hands  of  a  clock  coincide? 

Solution.  Since  in  one  hour  the  hour  hand  moves  through  5 
minute-spaces,  and  the  minute  hand  through  60  minute-spaces,  the 
minute  hand  moves  12  times  as  fast  as  the  hour  hand,  and  in  moving 
through  12  minute-spaces  gains  11  minute-spaces. 

When  the  hour  hand  is  at  V,  the  minute  hand,  being  at  XII,  is  25 
minute-spaces  behind.  Since  to  gain  11  minute-spaces  the  minute 
hand  must  move  through  12  minute-spaces,  to  gain  1  minute-space  the 
minute  hand  must  pass  through  Tf  of  1  minute-space,  and  to  gain  25 
minute-spaces,  it  must  pass  through  25  X  {f,  or  27T3r  minute-spaces. 

Hence,  the  hands  coincide  when  the  minute  hand  has  moved 
through  27T3T  minute-spaces ;  that  is,  at  27T3T  min.  after  5  o'clock. 

2.  At  what  time  between  10  and  11  o'clock  do  the  hour 
and  minute  hands  of  a  watch  coincide  ? 

3.  At  what  time  between  1  and  2  o'clock  do  the  hour 
and  minute  hands  of  a  clock  coincide  ? 

4.  At  what  time  between  8  and  9  o'clock  are  the  hands 
of  a  clock  exactly  opposite  each  other  ? 

5.  At  what  time  between  11  and  12  o'clock  are  the  hands 
of  a  clock  exactly  opposite  each  other  ? 

6.  At  what  time  between  4  and  5  o'clock  are  the  hands 
of  a  clock  exactly  opposite  each  other  ? 

7.  At  what  time  between  2  and  3  o'clock  do  the  hands 
of  a  clock  make  right  angles  with  each  other  ? 

8.  At  what  times  between  6  and  7  o'clock  do  the  hands 
of  a  watch  make  right  angles  with  each  other  ? 

9.  At  what  time  between  7  and  8  o'clock  do  the  hands 
of  a  watch  make  an  angle  of  120°  with  each  other  ? 

10.  At  what  time  between  12  and  1  o'clock  do  the  hands 
of  a  watch  make  an  angle  of  60°  with  each  other  ? 


PROBLEMS. 


207 


Bills. 

373.  Bills.  A  bill  is  a  written  statement  of  goods  sold, 
or  services  rendered,  giving  the  price  of  each  item  and  the 
total  cost,  as  well  as  the  date  of  each  transaction,  and  the 
names  of  the  parties  concerned. 

The  party  who  owes  is  called  the  Debtor,  and  the  party 
to  whom  a  debt  is  owed  is  called  the  Creditor. 


Mr.  George  Brown, 


(Specimen  of  a  Bill.) 

Boston,  Mass. ,  March  9,  1896. 

Bought  of  JAMES  BATES. 


1896 

Jan. 

15 

10  lb.  Coffee 

@  35/ 

$3 

50 

22 

11  lb.  Lard 

@     9? 

0 

99 

Feb. 

5 

25  lb.  Sugar 

@     5? 

1 

25 

12 

2  lb.  Tea 

@  65/ 

1 

$7 

30 
04 

(Specimen  of  a  Receipted  Bill.) 

Boston,  Mass.,  March  17,  1896. 
Mr.  John  Jones, 

To  JAMES  BROWN,  Dr. 


1896 

Jan. 

22 

To  40  t.  Coal 

@  $4-75 

$190 

00 

29 

To  20  cd.  Wood 

@    3.25 

65 

00 

$255 

00 

Cr. 

Jan. 

29 

By  JfO  bbl.  Apples 

@  $3.50 

140 

00 

Feb. 

10 

By  50  bu.  Potatoes 

@     0.80 

40 

00 

180 

00 

Balance  due 

$75 

00 

1896,  March  26. 


Received  payment, 

James  Brown. 


208  PROBLEMS. 

Exercise  96. 

Make  out  receipted  bills  for  the  following  accounts,  sup- 
plying dates  : 

1 .  James  Hardy  bought  of  C.  H.  Mills  275  bbl.  flour,  at 
$6.75 ;  324  bbl.  flour,  at  $6.25 ;  300  bu.  potatoes,  at  48 
cents  ;  1578  lb.  butter,  at  32  cents ;  2000  bbl.  apples,  at 
$1.25  ;  a  car-load  (20,000  lb.)  of  oats,  at  42  cents  a  bushel ; 
a  car-load  (28,575  lb.)  of  corn,  at  55  cents  a  bushel. 

2.  James  Harlow  bought  of  John  Pike  12  bales,  480  lb. 
each,  Texas  cotton,  at  9£  cents ;  7  bales,  502  lb.  each, 
upland,  at  10 J  cents  ;  3  bales,  492  lb.  each,  low  middling, 
at  9}  cents ;  18  bales,  490  lb.  each,  good  ordinary,  at 
9  cents. 

3.  Richard  Rowe  bought  of  John  Doe  125  lb.  sugar,  at 
5  cents  ;  1  bag  coffee,  115  lb.,  at  32  cents  a  pound ;  25  gal. 
molasses,  at  38  cents ;  8  lb.  Japan  tea,  at  92  cents ;  28  lb. 
crackers,  at  8  cents ;   2  bbl.  flour,  at  $7.50. 

4.  William  Litchfield  bought  of  John  Garvin  8  bags 
cracked  corn,  at  75  cents  ;  4  bags  oats,  at  80  cents ;  16  lb. 
sweet  potatoes,  at  3£  cents ;  2  bu.  potatoes,  at  $1.10  ; 
100  lb.  wire  nails,  at  2£  cents  ;   5  lb.  coffee,  at  35  cents. 

5.  Amos  Tuck  sold  to  Aaron  Young  11  lb.  ham  at 
15  cents,  22  lb.  beefsteak  at  24  cents,  18  lb.  mutton  at 
13  cents,  14  lb.  veal  at  11  cents ;  and  took  in  exchange 
5  dozen  eggs  at  18  cents,  15  lb.  butter  at  26  cents,  9  bu. 
potatoes  at  40  cents,  and  2  bbl.  apples  at  $1.35. 

6.  W.  G.  Fernald  sold  to  John  Waldron  35  lb.  sugar  at 
5  cents,  18  lb.  coffee  at  35  cents,  20  lb.  rice  at  8  cents, 
4  tons  hay  at  $15.75,  3  cords  pine  wood  at  $2.75,  4  cords 
hard  wood  at  $3.50,  8  tons  furnace  coal  at  $6.75,  5  tons 
stove  coal  at  $7.25,  8  rolls  wall  paper  at  35  cents ;  and 
took  in  exchange  25  bbl.  apples  at  $1.15,  32  bu.  pears  at 
60  cents,  and  42  bu.  blueberries  at  8  cents  a  quart. 


PROBLEMS.  209 

7.  C.  A.  Colton  bought  of  Green,  Fisk  &  Co.  4  doz. 
No.  7  teakettles,  at  85  cents  each ;  2  safety  ash  barrels, 
at  $2.50 ;  3  doz.  common  scrapers,  at  50  cents  a  dozen  ; 
8  eagle  shovels,  at  10  cents ;  -J-  doz.  8  by  12  black  registers, 
at  $  1.50  each  ;  -J  doz.  spice  boxes,  at  55  cents  each  ;  -J-  doz. 
14-qt.  dish  pans,  at  $ 6.00  a  dozen  ;  2  doz.  common  stove 
lifters,  at  50  cents  a  dozen ;  -J-  doz.  12  by  14  drip  pans, 
at  $4.00  a  dozen  ;  -J-  gross  retinned  teaspoons,  at  25  cents 
a  dozen  ;    1  doz.  ash  sifters  at  $1.00  each. 

8.  E.  M.  Hanson  bought  of  W.  F.  Fox  &  Co.  2  bbl. 
flour,  at  $5.75  ;  £  bbl.  fine  sugar,  153  lb.,  at  $4.81  a 
cwt.;  25  lb.  coffee,  at  33  cents;  3  lb.  Oolong  tea,  at 
50  cents  ;  15  pint  bottles  olives,  at  25  cents  ;  2  boxes 
graham  wafers,  at  40  cents  ;  \  doz.  cans  tomatoes,  at  $1.20 
a  dozen ;  \  doz.  cans  J.  H.  F.  peaches,  at  $3.50  a  dozen  ; 
4  Ferris  hams,  48  lb.,  at  12^-  cents  a  pound  ;  6  strips  Ferris 
bacon,  19  lb.  9  oz.,  at  13  cents  a  pound;  3  lb.  rice,  at  9 
cents  ;  3  lb.  tapioca,  at  5  cents  ;  40  lb.  rye  meal,  at  2-J- 
cents  ;  5  lb.  boneless  codfish,  at  14  cents ;  \  doz.  cans 
plums,  at  $2.90  a  dozen. 

9.  G.  B.  Cook  bought  of  Gray,  Higginson  &  Co.  1 
No.  8-20  Glenwood  B  range,  at  $35.00 ;  1  No.  12  Rock- 
ford  heater,  at  $20.00 ;  4  lb.  Eng.  stovepipe,  at  15  cents ; 
3  lb.  Rus.  stovepipe,  at  25  cents  ;  8  lb.  sheet  zinc,  at 
8-  cents  ;  1  stove  board,  at  $2.00 ;  1  set  kitchen  knives 
and  forks,  at  $1.50  ;  2  wash  tubs,  at  85  cents  ;  1  wash- 
board, at  25  cents;  1  set  Mrs.  Potts'  nickel  sad-irons,  at 
75  cents  ;  2  milk  cans,  at  35  cents  ;  1  hand  lamp  com- 
plete, at  30  cents  ;  1  stand  lamp,  at  $3.50 ;  1  granite  iron 
washbowl,  at  50  cents  ;  1  tea  canister  and  1  coffee  can- 
ister, at  20  cents  each  ;  1  carving  knife  and  fork,  at  $2.00 ; 
1  corn  popper,  at  25  cents  ;  1  rolling-pin,  at  20  cents ;  2 
8-qt.  porcelain  kettles,  at  70  cents  ;  1  granite  iron  coffee- 
pot, at  75  cents. 


CHAPTER  X. 


374. 


METBIC  AND  COMMON  SYSTEMS. 
Table  of  Equivalents. 


Length. 


Meter 
Kilometer 


_  (  39.37043  in.,  or 
~  t  1.09362  yd. 
=  0.62138  mi. 


Inch  =2.63998"" 
Yard  =  0.91439™. 
Mile  =  1.60933k" 


Surface. 


(  1550.031  sq.  in.,  or 
Sq'meter=  I  1.19601  sq.  yd. 
Hektar      =  2.47110  A. 


Sq.  inch  =  6.45148vm 
Sq.  yard  =  0.8361 1*"'. 
Acre        =  0.40468h». 


Volume. 


Cu.  centimeter  ss  0.06103  cu.  in. 
Cu.  meter  =  1.30799  cu.  yd. 

Ster  =  0.27590  cd. 


Cu.  inch  =  16.38662ccm. 
Cu.  yard  =  0.76453<*">. 
Cord        =    3.62446'*. 


Capacity. 


Liter  = 


1.05671  liquid  qt.,  or 
0.90810  dry  qt. 


Liquid  quart  =  0.946331. 
Dry  quart      =  1.101191. 


Weight. 


Milligram  =    0.016432  gr. 
Gram  =  15.43235  gr. 

Kilogram    =    2.20462  lb.  av. 
Metric  ton  =    2204.62  lb.  av. 


Grain  =    0.06480k. 

Ounce  av.  =  28.349548. 
Ounce  troy  =  31.103508. 
Pound  av.    =    0.45359k*. 


METRIC   AND    COMMON    SYSTEMS. 


211 


375. 


Table  of  Approximate  Equivalents. 


Meter 

=  1.1  yd. 

Yard 

=  0.9m. 

Kilometer 

=  I  mi. 

Mile 

=  1.6km. 

Sq.  meter 
Hektar 

=  li  sq.  yd. 
=  21  A. 

Sq.  yard 
Acre 

=  2ha# 

Cu.  centimetei 

=  tl  cu.  in. 

Cu.  inch 

—  \Qcemm 

Cu.  meter 
Ster 

=  1.3  cu.  yd. 

=  A  Cd. 

Cu.  yard 
Cord 

z=  1  ocbm 
T3 

=  8|  sters. 

Liter 
Hektoliter 

-/ixViiq-qt^or 

lTVdryqt. 
=  2|bu. 

Liquid  quart 
Dry  quart 
Bushel 

=  ||  liter. 
=  1^  liters. 

Gram 

=  15|  gr. 

Pound  av. 

-**■ 

Kilogram 

=  2i  lb.  av. 

Pound  troy 

=  #* 

Note.  These  tables  are  given  for  reference,  and  are  not  to  be 
committed  to  memory.  It  may  be  well,  however,  to  remember  that 
a  meter  is  39.37  in.;  a  liter  is  1.0567  liquid  qt.,  or  0.908  dry  qt.; 
and  that  a  gram  is  15.432  gr.;  as  by  these  equivalents  all  measures 
expressed  in  one  system  may  be  converted  into  the  corresponding 
measures  of  the  other  system. 


Exercise  97. 

In  the  following  problems  take  the  equivalents  from  the  Table  of 
Equivalents,  using  three  places  of  decimals,  or  four  when  the  first 
decimal  figure  is  zero,  and  add  one  to  the  last  decimal  figure  when  the 
next  is  5  or  more. 

1.  Reduce  25.55kg  to  pounds  avoirdupois. 

Solution.  Since  lke  =  2.205  lb.,  26.55k&  =  25.55  X  2.205  lb.,  or 
56.33775  lb.,  that  is,  56  lb.  5.4  oz. 

2.  Reduce  5  sq.  yd.  6  sq.  ft.  108  sq.  in.  to  square  meters. 

Solution.  5  sq.  yd.  6  sq.  ft.  108  sq.  in.  =  5.75  sq.  yd.  Since 
I  sq.  yd.  =  0.8361",  5.75  sq.  yd.  =  5.75  X  0.836*m;  that,  is  4.8074m. 

3.  Reduce  24  gal.  to  liters. 

4.  Reduce  10  lb.  troy  to  kilograms. 

5.  Reduce  50.5  cu.  yd.  to  cubic  meters. 


212  METRIC    AND   COMMON    SYSTEMS. 

6.  Reduce  69^^  mi.  to  kilometers. 

7.  Reduce  12  A.  12  sq.  rd.  to  hektars. 

8.  Reduce  10  cd.  to  sters. 

9.  Reduce  4  cwt.  24  lb.  to  kilograms. 

10.  Reduce  25  bu.  2  pk.  to  hektoliters. 

11.  Express  15km  in  the  common  system. 

12.  Express  3^  in  the  common  system. 

13.  Express  12.125cbm  in  the  common  system. 

14.  Express  101. 251  in  the  common  system. 

15.  Reduce  20.25M  to  liquid  quarts  ;  to  dry  quarts. 

16.  Express  5kg  in  troy  weight. 

17.  Express  248t  in  the  common  system. 

18.  Express  62.5qm  in  the  common  system. 

19.  Express  1001kg  in  avoirdupois  weight. 

20.  Express  42  A.  100  sq.  rd.  in  the  metric  system. 

21.  Find  in  acres,  etc.,  the  area  of  a  rectangular  field 
if  it  is  100m  long  and  75m  broad. 

22.  Find  the  number  of  cubic  meters  in  a  rectangular 
box  2  yd.  long,  3  ft.  wide,  and  2£  ft.  deep. 

23.  Find  the  number  of  cubic  yards  in  a  rectangular 
box  2m  long,  75cm  wide,  and  50cm  deep. 

24.  If  a  man  walks  75m  a  minute,  what  is  his  rate  in 
miles  per  hour? 

25.  If  a  cubic  centimeter  of  cast  iron  weighs  7.113g, 
how  many  pounds  does  a  cubic  foot  weigh  ? 

26.  How  many  steps  2  ft.  6  in.  long  will  a  man  take  in 
walking  a  kilometer  ? 

27.  Find  the  value  of  a  carboy  (17  qt.)  of  sulphuric 
acid,  specific  gravity  1.841,  at  4|  cents  a  kilogram. 

28.  Find  the  value  of  a  carboy  (17^)  of  nitric  acid, 
specific  gravity  1.451,  at  15  cents  a  pound. 

29.  If  the  specific  gravity  of  sea  water  is  1.026,  and  that 
of  olive  oil  is  0.915,  what  is  the  weight  of  a  hektoliter  of 
each  in  pounds  and  in  kilograms? 


METRIC    AND    COMMON    SYSTEMS.  213 

30.  Find  the  weight  in  pounds  and  in  kilograms  of  3l£ 
gal.  of  the  best  alcohol,  specific  gravity  0.792. 

31.  Find  the  weight  in  pounds  and  in  kilograms  of  the 
air,  specific  gravity  0.00129206,  in  a  room  7m  long,  5m  wide, 
and  3.5m  high. 

32.  Find  the  weight  in  pounds  and  in  kilograms  of  the 
air,  specific  gravity  0.00129206,  in  a  room  23  ft.  long,  16 
ft.  wide,  and  10  ft.  high. 

33.  What  is  the  lifting  force  in  kilograms  and  in  pounds 
of  a  balloon  that  weighs  2ks,  and  contains  10,000*  of  hydro- 
gen gas,  specific  gravity  0.00008929  ? 

Note.  The  lifting  force  is  the  weight  of  the  air  displaced  by  the 
balloon  diminished  by  the  weight  of  the  hydrogen  and  the  balloon. 

34.  What  is  the  value  at  $4.50  a  cord  of  a  pile  of  wood 
1.2m  wide,  7m  long,  and  2m  high  ? 

35.  How  many  miles  will  a  train  run  in  1  hr.  28  min. 
21  sec,  at  the  rate  of  50km  an  hour  ? 

36.  Find  the  time  it  takes  a  train  to  run  31  mi.  180  yd. 
at  the  rate  of  1  min.  25  sec.  per  kilometer. 

37.  What  is  the  weight  of  12  cu.  yd.  16  cu.  ft.  720  cu. 
in.  of  earth,  if  a  cubic  meter  weighs  1  t.  17  cwt.? 

38.  Find  the  weight  in  grams  of  a  liter  of  mercury,  if  a 
cubic  inch  weighs  0.4925  of  a  pound  avoirdupois. 

39.  How  many  yards  of  cloth,  at  $3.12^  a  meter,  should 
be  given  in  exchange  for  15m  at  $2.75  a  yard  ? 

40.  If  a  wine  merchant  buys  3W  of  wine  for  1600  francs, 
what  does  a  gallon  cost  him  in  United  States  money,  if 
25  francs  are  equivalent  to  $4,825  ? 

41.  A  mill  wheel  is  turned  by  a  stream  of  water  running 
at  the  rate  of  a  yard  a  second  in  a  channel  5  ft.  wide  and 
9  in.  deep.  Find  the  weight  in  metric  tons  and  in  tons 
avoirdupois  of  the  water  supplied  in  12  hr.,  if  a  cubic  foot 
of  water  weighs  1000  oz. 


214  METRIC   AND   COMMON   SYSTEMS. 

Exercise  98. 

In  the  following  problems  take  the  equivalents  from  the  Table  of 
Approximate  Equivalents,  and  use  *f-  for  3.1410. 

1.  When  water  is  heated  from  the  freezing  point  to 
the  boiling  point  it  expands  J±  in  volume.  Find  in  kilo- 
grams the  weight  of  a  cubic  foot  of  water  at  the  freezing 
point  and  at  the  boiling  point. 

2.  A  circular  plate  of  lead  8  in.  in  diameter  and  2  in. 
thick  is  changed  without  loss  into  spherical  shot  each 
1.25mm  in  radius.     How  many  shot  does  it  make  ? 

3.  If  |  of  a  yard  of  velvet  costs  $3,  how  many  francs 
will  f  of  a  meter  cost  ? 

4.  Water  expands  ^  in  freezing,  and  a  floating  body 
displaces  an  amount  of  water  equal  in  weight  to  the  body. 
What  is  the  volume  in  cubic  meters  and  the  weight  in 
metric  tons  of  an  iceberg  floating  in  the  ocean,  if  the  spe- 
cific gravity  of  sea  water  is  1.026,  and  the  part  of  the  iceberg 
above  the  water  is  a  rectangular  solid  200  ft.  long,  60  ft. 
wide,  and  12  ft.  high? 

5.  How  many  hektoliters  of  wheat  will  a  rectangular 
bin  hold  14  ft.  long,  10  ft.  wide,  and  6  ft.  high  ? 

6.  How  many  hektoliters  of  water  will  a  cylindrical 
stand-pipe  hold  70  ft.  high  and  35  ft.  in  diameter  ? 

7.  How  many  bushels  of  wheat  will  a  rectangular  bin 
hold  4m  long,  3m  wide,  and  2.5™  high  ? 

8.  How  many  gallons  of  water  in  a  well  1.2m  in  diam- 
eter, if  the  depth  of  the  water  is  2m  ? 

9.  If  1  lb.  troy  of  silver  is  worth  $6.20,  what  is  the 
value  of  a  lump  of  silver  weighing  2.64kg  ? 

10.  A  pound  of  brass  contains  3.3  cu.  in.,  and  a  pound 
of  antimony  contains  6.27  cu.  in.  Find  the  weight  in  kilo- 
grams of  a  mass  of  313^  cu.  in.  that  contains  equal  volumes 
of  the  two  metals. 


METRIC    AND    COMMON    SYSTEMS.  215 

11.  If  2  cu.  in.  of  mercury  weighs  1  lb.,  and  100  cu.  in. 
of  air  weighs  31  gr.,  how  many  kilometers  high  must  a 
column  of  air  be  to  weigh  as  much  as  a  column  of  mercury 
29.388  in.  high,  standing  on  a  base  of  the  same  area  ? 

12.  If  a  sprinter  can  run  0.00645  of  a  mile  in  1.08  sec, 
how  many  meters  can  he  run  in  a  second  ?  How  many 
seconds  will  it  take  him  to  run  100m? 

13.  Two  trains  going  in  opposite  directions  pass  each 
other  in  3^  sec.  If  their  lengths  are  260  ft.  and  200  ft., 
respectively,  and  the  first  train  is  going  at  the  rate  of  80km 
an  hour,  what  is  the  rate  of  the  second  train  ? 

14.  If  a  cubic  inch  of  water  converted  into  steam  will 
produce  mechanical  force  sufficient  to  raise  a  weight  of 
2200  lb.  one  foot  high,  how  many  meters  high  would  the 
conversion  into  steam  of  a  cubic  centimeter  of  water  raise 
a  weight  of  one  kilogram  ? 

15.  If  a  man  takes  100  steps  of  0.7m  each  in  a  minute, 
how  long  will  it  take  him  to  walk  a  distance  of  28km  ? 

16.  A  lot  of  land  containing  63a  21ca,  worth  $0.35  a 
square  yard,  is  exchanged  for  a  second  lot  containing  V3* 
5a.     What  is  the  cost  per  ar  of  the  second  lot  ? 

17.  Light  travels  in  8  min.  13  sec.  from  the  sun  to  the 
earth,  153,624,000km.  What  is  the  velocity  of  light  in 
miles  per  second  ? 

18.  How  many  square  feet  of  surface  has  a  rectangular 
table  that  is  l.lm  long  and  0.85m  wide  ? 

19.  How  many  square  meters  of  surface  has  a  circular 
table  that  is  3£  ft.  in  diameter  ? 

20.  If  sound  travels  340m  a  second,  how  many  feet  dis- 
tant is  a  cannon  from  a  man  who  hears  the  report  13  sec. 
after  he  sees  the  flash  ? 

21.  How  many  square  meters  of  zinc  will  be  required 
to  line  a  rectangular  cistern,  open  at  the  top,  12  ft.  long, 
10  ft.  wide,  and  8  ft.  deep  ? 


216  METRIC    AND   COMMON    SYSTEMS. 

22.  A  rectangular  tank  is  3m  long,  2£m  wide,  and  l£m 
high,  external  measurement.  If  its  sides  are  0.1m  thick, 
how  many  gallons  of  water  will  the  tank  hold  ? 

23.  If  a  cube  of  pine  wood  11.2cm  on  an  edge  weighs 
2  lb.,  what  is  the  specific  gravity  of  the  pine  ? 

24.  Find  in  kilograms  the  weight  of  water  a  cubical  cis- 
tern will  hold,  6  ft.  on  an  edge. 

25.  Rain  has  fallen  to  the  depth  of  half  an  inch.  How 
many  cubic  meters  of  water  lias  fallen  on  an  acre  of  land  ? 

26.  How  many  centimeters  will  the  water  sink  in  a 
cylindrical  cistern  7  ft.  in  diameter,  if  310  gallons  of  water 
is  pumped  out  ? 

27.  How  many  square  yards  of  tin  are  required  to  cover 
the  roof  of  a  hemispherical  dome  12m  in  diameter  ? 

28.  If  a  cubic  inch  of  iron  weighs  4£  oz.,  what  is  the 
weight  in  kilograms  of  an  iron  ball  10cm  in  diameter  ? 

29.  If  a  cubic  inch  of  lead  weighs  7  oz.,  what  is  the 
weight  in  kilograms  of  a  lead  pipe  3m  long,  6cm  in  external 
diameter,  if  the  pipe  is  lcm  thick  ? 

30.  Find  the  cost  at  $7.25  per  meter  of  building  a  wall 
around  a  rectangular  garden  90  ft.  long  and  55  ft.  wide. 

31.  The  minute  hand  of  a  clock  is  0.5m  long.  How 
many  feet  does  its  point  move  in  an  hour  ? 

32.  A  spherical  shot  3  in.  in  diameter  is  melted  and  then 
cast  into  a  cylinder  9cm  in  diameter.  What  is  the  height 
in  centimeters  of  this  cylinder  ? 

33.  What  is  the  cost  at  $18  per  1000  ft.  board  measure 
of  4  beams,  each  4.5m  long,  7.5cm  wide,  and  5cm  thick  ? 

34.  The  radius  of  a  cylindrical  roller  is  0.4m  and  its 
length  is  2.15m.     Find  its  volume  in  cubic  feet. 

35.  A  cylindrical  cistern,  the  circumference  of  whose 
base  is  2.2m,  and  whose  depth  is  2.1m,  is  four  fifths  filled 
with  water.  Find  in  gallons  the  volume  of  the  water,  and 
in  pounds  the  weight  of  the  water. 


CHAPTER    XI. 

RATIO  AND  PROPORTION. 

376.  Ratio.  The  relative  magnitude  of  two  numbers 
is  called  their  ratio,  when  expressed  by  the  fraction  that 
has  the  first  number  for  its  numerator  and  the  second 
number  for  its  denominator. 

Thus,  the  ratio  of  2  to  3  is  expressed  by  the  fraction  f . 

377.  Antecedent  and  Consequent.  The  terms  of  this 
fraction  are  called  the  terms  of  the  ratio.  The  first  term 
of  a  ratio  is  called  the  antecedent;  the  second  term,  the 
consequent. 

Thus,  in  the  ratio  of  2  to  3,  commonly  written  2  :3,  the  first  term 
2  is  the  antecedent,  and  the  second  term  3  is  the  consequent. 

378.  If  both  terms  of  a  ratio  are  multiplied  or  both 
divided  by  the  same  number,  the  value  of  the  ratio  is  not 
changed. 

Thus,  if  the  ratio  2\  :  3^  is  multiplied  by  6,  the  resulting  ratio  is 

21 
15  :  20,  and  the  ratio  2\  :  3|  is  equal  to  15  :  20 ;  for  ;r|  =  ]§.     Since 

|f  reduced  to  its  lowest  terms  =  £,  the  simplest  expression  for  the 
ratio  of  2| :  S\  is  3  : 4. 

379.  If  the  numerator  and  denominator  of  a  fraction 
are  interchanged,  the  fraction  is  said  to  be  inverted;  like- 
wise, if  the  antecedent  and  consequent  of  a  ratio  are  inter- 
changed, the  resulting  ratio  is  the  inverse  of  the  given  ratio. 

Thus,  if  the  fraction  £  is  inverted  the  resulting  fraction  is  f ;  and 
the  inverse  of  the  ratio  3  :  4  is  4  : 3. 


218  RATIO   AND   PROPORTION. 

380.  If  two  quantities  are  expressed  in  the  same  unity 
their  ratio  is  the  same  as  the  ratio  of  the  two  numbers  by 
which  they  are  expressed. 

Thus,  the  quantity  $5  is  the  same  fraction  of  $11  as  5  is  of  11 ;  and, 
therefore,  the  ratio  $5  :  $11  equals  the  ratio  6:11. 

381.  Since  ratio  is  simply  relative  magnitude,  two  quan- 
tities different  in  kind  cannot  form  the  terms  of  a  ratio ; 
and  two  quantities  the  same  in  kind  must  be  expressed  in 
a  common  unit  before  they  can  form  the  terms  of  a  ratio. 

Thus,  no  ratio  exists  between  5  tons  and  30  days ;  and  the  ratio  of 
5  tons  to  3000  pounds  can  be  expressed  only  when  both  quantities  are 
written  as  tons  or  as  pounds. 

382.  Since  ratios  are  mere  numbers,  they  may  be  com- 
pared. 

383.  Example.  Which  is  the  greater  ratio,  5  : 7  or 
12  :  18  ? 

Solution.     5  :  7  =  f,  and  12  :  18  =  jf  =  § . 

Now,  »  =  #,  and$'=  \\. 

As  \\  is  greater  than  \\ , 

the  ratio  5  :  7  is  greater  than  the  ratio  12  :  18. 

Exercise  99. 
Which  is  the  greater  ratio  : 

1.  5:8  or  6:9?  5.    10  cwt. :  15  cwt.  or  $7  :  $9  ? 

2.  7  :  10  or  9  :  12  ?         6.    5  dy. :  7  dy.  or  8  ft.  :  11  ft.? 
3.8:9  or  10  :  12  ?         7.    9  yd. :  6  yd.  or  5  :  3  ? 

4.  6  :  12  or  8  :  14  ?         8.    §  lb. :  \  lb.  or  £  yd.  :  f  yd.? 

9.  Find  the  ratio  of  3  dry  quarts  to  2  pecks. 

10.  Find  the  ratio  of  2500  lb.  to  1  ton. 

11.  Find  the  ratio  of  a  rectangular  field  16  rd.  long,  14 
rd.  wide  to  a  rectangular  field  14  rd.  long,  12  rd.  wide. 

12.  Find  the  ratio  of  a  circle  1  in.  in  diameter  to  a 
circle  1  in.  in  radius. 


RATIO   AND   PROPORTION.  219 

384.  When  two  ratios  are  equal  the  four  terms  are  said 
to  be  in  proportion,  and  are  called  proportionals. 

Thus,  5,  3,  15,  9  are  proportionals  ;  for  f  =  -L5-. 

385.  Proportion.  An  expression  of  equality  between 
two  ratios  is  called  a  proportion. 

A  proportion  is  written  by  putting  the  sign  of  equality 
or  a  double  colon  between  the  ratios. 

Thus,  5  :  3  =  15  :  9,  or  5  : 3  ::  15  :  9,  means,  and  is  read,  the  ratio  of 
5  to  3  is  equal  to  the  ratio  of  15  to  9. 

386.  Means  and  Extremes.  The  first  and  last  terms 
of  a  proportion  are  called  the  extremes,  and  the  two  middle 
terms  are  called  the  means. 

387.  Test  of  a  Proportion.  When  four  numbers  are  in 
proportion,  the  product  of  the  extremes  is  equal  to  the  product 
of  the  means. 

This  is  seen  to  be  true  by  expressing  the  ratios  in  the  form  of  frac- 
tions, and  multiplying  both  by  the  product  of  the  denominators. 

Thus,  the  proportion  5 :  3  =  15  :  9  may  be  written  f  =  -1/- ;  and  if 
both  are  multiplied  by  3  X  9,  the  result  is  5  X  9  =  3  X  15. 

388.  Either  extreme,  therefore,  is  equal  to  the  product 
of  the  means  divided  by  the  other  extreme ;  and  either 
mean  is  equal  to  the  product  of  the  extremes  divided  by 
the  other  mean.  Hence,  if  three  terms  of  a  proportion  are 
given,  the  fourth  may  be  found. 

389.  Examples.  1.  Find  the  missing  term  of  the  pro- 
portion 18  :  32  =  45:?. 

32  X  45      _ 


Solution. 


18 


2.    Find  the  missing  term  of  20  :  24  =  ?  :  30. 

Solution.  — — —  =  25. 

24 


220  RATIO    AND   PROPORTION. 

Exercise  100. 
Find  the  missing  term  of  : 

1.  24: 18::  16:?.  6.  18:?::  32:  45. 

2.  35:?::  15:  21.  7.  ?:12::5:18. 

3.  45:  40::?:  32.  8.  8:17  ::?:119. 

4.  30:  27::  40:?.  9.  9: 16::  12:?. 

5.  ?:36::4  :  3.  10.  17:  3::?:  12. 

390.  Rule  of  Three.  When  three  terms  of  a  proportion 
are  given,  the  method  of  finding  the  fourth  term  is  called 
the  Rule  of  Three. 

It  is  usual  to  arrange  the  quantities  (that  is,  to  state  the 
question),  so  that  the  quantity  required  for  the  answer 
may  be  the  fourth  term.  Hence,  the  quantity  which  come- 
sponds  to  that  of  the  required  answer  is  the  third  term. 

391.  Examples.  1.  If  5  tons  of  hay  cost  $87.50,  what 
will  21  tons  cost  ? 

Solution.     Since  cost  is  required,  $87.50  is  the  third  term. 

Since  21  tons  will  cost  more  than  5  tons,  21  tons  is  the  second  terra 
and  5  tons  the  first  term. 

That  is,  6  t.  :  21  t.  ::  $87.50  :  ?. 

A  difficulty  presents  itself  here,  inasmuch  as  no  meaning  can  be 
given  to  the  product  of  the  means  ($87.50  multiplied  by  21 1.).  Since, 
however,  the  ratio  of  5  t.  :  21  t.  =  the  ratio  of  5  :  21,  the  ratio  6:21 
may  be  substituted  for  the  ratio  6  t.  :  21  t. 

Then,  5:21  ::  $87.50:  ?. 

21  X  $87  50 
Therefore,  the  fourth  term  required  is — '- — ,  or  $367.60. 

2.  When  a  post  11.5  ft.  high  casts  a  shadow  on  level 
ground  17.4  ft.  long,  a  neighboring  steeple  casts  a  shadow 
63.7  yd.  long.     How  high  is  the  steeple  ? 

Solution.  Since  height  is  required,  the  height  11.5  ft.  is  the  third 
term. 


RATIO    AND   PROPORTION.  221 

Since  the  shadow  of  the  steeple  is  the  longer,  the  height  of  the 
steeple  must  be  the  greater ;  therefore,  the  second  term  must  be  the 
greater  of  the  two  remaining  quantities  expressed  in  the  same  unit. 
63.7  yd.  =  191.1  ft.     Therefore, 

Shadow.         Shadow.  Height.         Height. 

17.4  ft.  :  191.1  ft. ::  11.5  ft.  :  What  ? 
or,  17.4        :  191.1        ::  11.5  ft.  :  What  ? 

Hence,  the  height  of  the  steeple  is '■ — titt1 \  or  126.3  ft. 

392.   To  Solve  Problems  by  the  Rule  of  Three, 

Make  that  quantity  which  is  of  the  same  kind  as  the  re- 
quired answer  the  third  term. 

Make  the  numbers  by  which  the  two  remaining  quantities 
are  represented  when  expressed  in  the  same  unit  the  first 
and  second  terms. 

If,  from  the  nature  of  the  question,  the  answer  will  be 
greater  than  the  third  term,  make  the  greater  of  these  two 
numbers  the  second  term  ;  if  the  answer  will  be  smaller  than 
the  third  term,  make  the  smaller  of  these  numbers  the  second 
term,  and  the  larger  the  first  term. 

Divide  the  product  of  the  second  and  third  terms  by  the 
first  term,  and  the  quotient  will  be  the  answer  required. 

Exercise  101. 

1.  If  24  men  can  do  a  piece  of  work  in  14  days,  how 
long  will  it  take  21  men  to  do  it  ? 

2.  A  well  is  dug  in  13  days  of  9  hours  each.  How  many 
days  of  10  hours  each  wonld  it  have  taken  ? 

3.  A  man  who  steps  2  ft.  5  in.  takes  2480  steps  in 
walking  a  certain  distance.  How  many  steps  of  2  ft.  7  in. 
will  be  required  for  the  same  distance  ? 

4.  If  T%  of  a  ton  of  hay  costs  $6,  what  will  7$  cwt. 
cost,  at  the  same  rate  ? 


222  RATIO    AND   PROPORTION. 

5.  If  42  yd.  of  carpet  2  ft.  3  in.  wide  are  required  for 
a  room,  how  many  yards  of  carpet  2  ft.  4  in.  wide  will  be 
required  ? 

6.  A  court  was  paved  with  950  stones,  each  containing 
lg  sq.  ft.,  and  is  repaved  with  836  stones  of  a  uniform  size. 
Find  the  surface  of  each. 

7.  If  a  train,  at  the  rate  of  *fc  of  a  mile  per  minute, 
requires  3£  hours  to  make  a  certain  distance,  how  long  will 
it  require  at  the  rate  of  *fo  of  a  mile  a  minute  ? 

8.  When  a  post  4  ft.  8  in.  high  casts  a  shadow  7  ft. 
3  in.  long,  how  long  a  shadow  will  a  post  11  ft.  high  cast  ? 

9.  When  a  post  5  ft.  7  in.  high  casts  a  shadow  8  ft.  5 
in.  long,  how  high  is  a  steeple  that  casts  a  shadow  of  202  ft.  ? 

10.  If  4  men  can  mow  a  certain  field  in  10  hours,  how 
many  men  will  it  take  to  mow  it  in  5  hours  ? 

11.  If  a  tap  discharging  4  gal.  a  minute  empties  a  cis- 
tern in  3  hours,  how  long  will  it  take  a  tap  discharging  7 
gal.  a  minute  to  empty  it  ? 

12.  If  a  pipe  discharging  3  gal.  1  pt.  a  minute  fills  a  tub 
in  4  min.  20  sec,  how  long  will  it  take  a  pipe  discharging 
83  qt.  a  minute  to  fill  it  ? 

13.  If  both  pipes  of  Ex.  12  discharge  at  the  same  time 
into  the  tub,  how  long  will  it  take  to  fill  it  ? 

14.  How  long  will  it  take  to  fill  a  cistern  of  165  gal.  by 
a  pipe  that  fills  one  of  120  gal.  in  7  min.  16  sec.  ? 

15.  If  a  ship  sails  1800  mi.  in  a  fortnight,  how  long  will 
it  take  to  make  a  voyage  of  5000  mi.  ? 

16.  The  wheels  of  a  carriage  are  6  ft.  9  in.  and  9  ft.  6 
in.,  respectively,  in  circumference.  How  many  times  will 
the  larger  turn  while  the  smaller  turns  3762  times  ? 

17.  If  &  of  a  ship  is  worth  $2167,  what  is  T7T  of  it 
worth  ? 

18.  What  is  the  weight  of  18  cu.  ft.  432  cu.  in.  of  stone, 
if  10  cu.  ft.  864  cu.  in.  of  the  stone  weighs  14  cwt.  7  lb.  ? 


RATIO   AND   PROPORTION.  223 

19.  If  280  lb.  of  flour  makes  360  lb.  of  bread,  how  many 
four-pound  loaves  can  be  made  from  1  cwt.  of  flour  ? 

20.  If  a  column  of  mercury  27.93  in.  high  weighs  0.76 
of  a  pound,  what  is  the  weight  of  a  column  of  mercury  of 
the  same  diameter  29.4  in.  high  ? 

21.  How  many  francs  will  pay  a  bill  of  £100,  when 
£42  10s.  8d.  is  equivalent  to  1090.98  francs  ? 

22.  What  is  the  weight  of  a  cube  of  stone  2  ft.  2  in.  on 
an  edge,  if  a  cube  1  ft.  4  in.  on  an  edge  weighs  537.6  lb.  ? 

23.  If  a  square  field  50  yd.  lOf  in.  on  a  side  is  worth 
$2710}^,  what  is  a  square  field  62  yd.  1  ft.  on  a  side 
worth  ? 

24.  A  gains  4  yd.  on  B  in  running  30  yd.  How  many 
yards  will  he  gain  while  B  is  running  97^  yd.  ? 

25.  If  10  cu.  in.  of  gold  weighs  as  much  as  193  cu.  in. 
of  water,  how  many  cubic  inches  are  there  in  a  nugget  of 
gold  that  weighs  as  much  as  a  cubic  foot  of  water  ? 

26.  If  a  garrison  of  1500  men  has  provisions  for  13 
months,  how  long  will  the  provisions  last  if  the  garrison  is 
reenforced  by  700  men  ? 

27.  If  a  tree  38  ft.  high  is  represented  by  a  drawing 
1J  in.  high,  what  height  on  the  same  scale  will  represent 
a  house  45  ft.  high  ? 

28.  If  a  country  630  mi.  long  is  represented  on  a  raised 
map  by  a  length  of  5^  ft.,  by  what  height  ought  a  moun- 
tain of  15,750  ft.  be  represented  on  the  map  ? 

29.  A  train  travels  \  of  a  mile  in  18  sec.  How  many 
miles  an  hour  does  it  travel  ? 

30.  If  4£  t.  of  coal  fill  a  bin  9  ft.  long,  5  ft.  broad,  5  ft. 
high,  how  many  cubic  feet  are  required  for  the  coal  of  a 
steamer  that  carries  coal  for  3  wk.  at  20  t.  a  day  ? 

31.  If  2  lb.  of  rosin  are  melted  with  5  oz.  of  mutton 
tallow  to  make  a  grafting  wax,  how  many  ounces  of  tallow 
will  20  oz.  of  the  wax  contain  ? 


224  RATIO   AND   PROPORTION. 

Compound  Proportion. 

393.  Compound  Ratio.  A  ratio  is  said  to  be  com- 
pounded of  two  or  more  given  ratios  when  it  is  expressed 
by  a  fraction  that  is  the  product  of  the  fractions  repre- 
senting the  given  ratios. 

Thus,  the  ratios  2  : 3  and  7:11  are  represented  by  the  fractions  \ 
and  T\  ;  and  the  ratio  14  :  33,  which  is  represented  by  ^f  (the  product 
of  $  and  /y),  is  said  to  be  compounded  of  the  ratios  2  :  3  and  7:11. 

394.  Compound  Proportion.  A  proportion  which  has 
one  of  its  ratios  a  compound  ratio  is  called  a  compound 
proportion. 

In  stating  problems  in  compound  proportion  the  quantity 
that  corresponds  to  the  answer  required  is  made  the  third 
term.  Each  pair  of  the  remaining  quantities  is  then  con- 
sidered separately  with  reference  to  the  answer  required. 

395.  Example.  If  4  men  mow  15  A.  in  5  dy.  of  14  hr., 
in  how  many  days  of  13  hr.  can  7  men  mow  19£  A.? 

As  the  answer  is  to  be  in  days,  we  make  5  dy.  the  third  term. 

I.  It  will  require  less  days  for  7  men  to  mow  16  A.  than  for  4  men. 
Therefore,  we  make  7  the  first  term,  and  4  the  second. 

II.  It  will  require  more  days  for  the  same  number  of  men  to  mow 
19J  A.  than  to  mow  15  A. 

Therefore,  we  make  16  the  first  term,  and  19^  the  second. 

III.  It  will  require  more  days  of  13  hr.  than  of  14  hr.  for  the  same 
number  of  men  to  mow  the  same  number  of  acres. 

Therefore,  we  make  13  the  first  term,  and  14  the  second. 
Hence,  the  statement  is 

7:4 

15  :  19.5  : :  5  dy.  :  ? 

13 :  14. 

Therefore,  the  fourth  term,  or  the  time  required,  is 


4  X  19.5  X  14  X  5  dy. 

7  X  15  X  13  *' 


RATIO   AND   PROPORTION.  225 


Exercise  102. 


1.  In  how  many  days  of  8  hr.  will  60  men  do  the 
same  work  that  24  men  can  do  in  15  dy.  of  10  hr.? 

2.  What  is  the  expense  of  covering  a  room  with  drug- 
get 4  ft.  wide,  at  91 J  cents  a  yard,  if  carpet  2  ft.  3  in.  wide 
for  the  room  costs  $70.50,  at  $1.37£  a  yard  ? 

3.  If  4418  tons  of  iron  ore  produce  $36,190  worth  of 
metal,  when  iron  is  at  $37.50  a  ton,  what  will  be  the 
value  of  the  iron  at  $47  a  ton  from  2275  tons  of  ore  ? 

4.  If  a  bar  of  iron  3£  ft.  long,  3  in.  wide,  and  2f  in. 
thick  weighs  93  lb.,  what  will  be  the  weight  of  a  bar  3f  ft. 
long,  4  in.  wide,  and  2-J-  in.  thick  ? 

5.  If  40  bu.  of  wheat  can  be  grown  on  the  same  area  as 
48  bu.  of  barley,  and  28  A.  produce  840  bu.  of  wheat,  how 
mucn  barley  will  38  A.  produce  ? 

6.  If  18  men  can  dig  a  trench  150  ft.  long,  6  ft.  broad, 
and  4  ft.  6  in.  deep  in  12  days,  in  how  many  days  will  16 
men  dig  a  trench  210  ft.  long,  5  ft.  broad,  and  4  ft.  deep  ? 

7.  A  book  of  810  pages,  40  lines  to  a  page,  and  60 
letters  to  a  line,  is  reprinted  in  pages  of  50  lines,  72 
letters  to  a  line.  How  many  pages  will  the  new  edition 
contain  ? 

8.  If  3280  42-lb.  shot  cost  $3000,  how  many  32-lb.  shot 
can  be  bought  for  $4200  ? 

9.  What  is  the  rate  of  wages,  if  12  men  earn  in  i0  dy. 
as  much  as  9  men  earn  in  14  dy.,  at  $1.50  a  day  ? 

10.  A  rectangular  reservoir  15  yd.  long  and  4  ft.  deep 
holds  32,500  gal.  What  quantity  of  water  will  it  hold  if 
its  length  is  increased  by  18  ft.  and  its  depth  by  1  ft.? 

11.  What  must  be  the  length  of  a  bar  of  silver  f  in. 
square  to  weigh  the  same  as  a  bar  of  gold  £  in.  square  and 
6£  in.  long,  if  the  weight  of  a  cubic  inch  of  silver  to  that 
of  a  cubic  inch  of  gold  is  in  the  ratio  47  :  88  ? 


226  RATIO   AND   PROPORTION. 

12.  How  far  can  A,  who  takes  3.1  ft.  each  step,  walk, 
while  B,  who  takes  2.3  ft.  each  step,  walks  220  yd.,  if  A 
takes  7  steps  while  B  takes  11  ? 

13.  If  6  hr.  are  needed  to  go  a  given  distance  at  a  given 
rate,  how  many  hours  are  needed  when  the  distance  is  dimin- 
ished by  one  fourth  and  the  rate  increased  by  one  half  ? 

14.  How  many  hours  a  day  must  5  men  work  to  mow  a 
field  in  8  dy.  that  7  men  can  mow  in  6  dy.  of  10  hr.? 

15.  If  a  bar  of  iron  10  ft.  6 J  in.  long,  3 J  in.  broad,  and 
3J  in.  thick  weighs  4  cwt.  20.21  lb.,  what  is  the  length  of  a 
bar  of  iron  that  weighs  a  long  ton,  if  its  breadth  and  thick- 
ness are  4 J  in.  and  4-J  in.,  respectively  ? 

16.  If  27  men  in  28  dy.  of  10  hr.  dig  a  trench  126  yd. 
long,  2£  yd.  broad,  1£  yd.  deep,  how  long  a  trench  2f  yd. 
broad  and  If  yd.  deep  will  56  men  dig  in  25  dy.  of  8£  hr.? 

17.  If  34k*  of  wool  makes  25m  of  cloth  0.6m  wide,  how 
long  a  piece  of  cloth  0.8m  wide  will  108.8^  of  wool  make  ? 

18.  If  an  oak  beam  5.40m  long,  0.63m  thick,  and  0.57nB 
wide  weighs  1469.25^,  what  is  the  weight  of  an  oak  beam 
4.87m  long,  0.58m  thick,  and  0.53m  wide  ? 

19.  A  certain  quantity  of  air  has  a  volume  of  195.5  cu.  ft. 
at  27.8°  C.     What  will  be  its  volume  at  100°  C? 

Note.  The  coefficient  of  expansion  of  a  body  is  the  increase  of  a 
unit  of  volume  of  the  body  when  the  temperature  is  increased  1°  C. 
The  coefficient  of  expansion  for  air  is  0.00367.  That  is,  a  cubic  foot 
of  air  at  0°  C.  occupies  at  1°  C.  1.00367  cu.  ft.,  if  the  pressure  remains 
the  same.  The  increase  in  volume  is  found  approximately  by  multi- 
plying the  increase  of  1°  by  the  number  of  degrees. 

20.  A  quantity  of  air  at  a  temperature  of  15.6°  C.  has  a 
volume  of  4  cu.  ft.  under  a  pressure  of  12  lb.  per  square 
inch.  What  will  be  its  volume  at  48.7°  C.  under  a  pres- 
sure of  14  lb.  per  square  inch  ? 

Note.  The  volumes  occupied  by  the  same  quantity  of  air,  the  tem- 
perature remaining  unchanged,  are  in  inverse  ratio  to  the  pressures. 


RATIO   AND   PROPORTION.  227 

Cause  and  Effect. 

396.  Problems  in  compound  proportion  are  readily 
solved  by  the  cause  and  effect  method. 

397.  The  cause  and  effect  method  depends  upon  the 
following  principle  : 

Like  causes  produce  like  effects;  and  the  ratio  between 
any  two  causes  equals  the  ratio  between  the  effects  produced. 

Note.  Examples  of  causes  are  men  at  work,  time,  and  goods 
bought  or  sold ;  examples  of  effects  are  work  done,  wages,  and  cost 
or  selling  price  of  goods. 

398.  Example.  If  4  men  mow  15  A.  in  5  dy.  of  14  hr., 
in  how  many  days  of  13  hr.  can  7  men  mow  19£  A.? 


Solution.        1st  cause.     2d  cause.  1st  effect.    2d  effect. 

4  men  >>     7  men 

5  dy.    I  :  ?  dy.    } 
14  hr.  J      13  hr. 


:  I  19.5  A. 


Since  the  product  of  the  means  equals  the  product  of  the  extremes, 

the  number  of  days  required  equals  the  product  of  the  extremes  divided 

by  the  product  of  the  remaining  means. 

v-      *  a  -a-    4  X  5  X  14  X  19.5       . 

Hence,  the  number  of  days  required  is —  =  4. 

7  X  18  X  15 

Compare  this  solution  with  the  solution  of  the  example  in  §  395. 


Exercise  103. 
Solve  the  following  problems  by  the  cause  and  effect  method. 

1.  If  a  man  can  mow  ^T  of  a  field  in  a  day,  how  long 
will  it  take  another  man  to  mow  £  of  a  field  5^  times  as 
large,  if  the  second  man  works  If  times  as  fast  as  the  first 
but  only  $•  as  many  hours  each  day  ? 

2.  If  4  men  or  7  boys  can  do  a  piece  of  work  in  6  days, 
how  long  will  it  take  6  men  and  9  boys  to  do  the  work  ? 


228  RATIO   AND   PROPORTION. 

3.  If  50  men  working  9  hr.  a  day  require  6  dy.  to  dig 
a  trench  100  yd.  long,  2  yd.  wide,  and  3  yd.  deep,  how 
many  men  working  10  hr.  a  day  for  9  dy.  will  be  required 
to  dig  a  trench  50  yd.  long,  6  yd.  wide,  and  5  yd.  deep,  in 
ground  twice  as  hard  to  dig  ? 

4.  If  12  men  in  9  dy.  can  harvest  40  A.  of  wheat,  how 
many  acres  can  16  men  harvest  in  3  dy.? 

5.  If  120  men  can  make  an  embankment  £  of  a  mile 
long,  30  yd.  wide,  and  7  yd.  high,  in  42  dy.,  how  many  men 
will  it  take  to  make  an  embankment  1000  yd.  long,  36  yd. 
wide,  and  22  ft.  high,  in  30  dy.? 

6.  If  7  women  in  8  dy.  of  11  hr.  each  can  make  22 
dozen  shirts,  in  how  many  days  of  10  hr.  each  can  12 
women  make  360  dozen  shirts  ? 

7.  Twenty-five  lamps  used  5  hr.  an  evening  for  40  dy. 
required  a  quantity  of  oil  that  cost  $4.25.  How  many 
lamps  used  4  hr.  an  evening  for  30  dy.  can  be  furnished 
with  oil  at  a  cost  of  $7.65  ? 

8.  If  8  horses  can  be  kept  12  dy.  for  a  certain  sum 
when  hay  is  worth  $15  a  ton,  how  many  days  can  6 
horses  be  kept  for  the  same  sum  when  hay  is  worth  $12 
a  ton  ? 

9.  Twenty  horses  working  14  wk.,  6  dy.  a  week  and 
8  hr.  a  day,  transport  the  output  of  a  mine  to  the  nearest 
wharf.  In  how  many  weeks  will  24  horses  do  the  same 
work,  if  they  work  5  dy.  a  week  and  7  hr.  a  day  ? 

10.  If  6  men  can  reap  a  field  of  rye  200  yd.  long  and 
150  yd.  wide  in  4  dy.  of  12  hr.  each,  in  how  many  days  of 
10  hr.  each  will  8  men  reap  a  field  300  yd.  long  and  250  yd. 
wide  ? 

11.  If  a  boy  can  do  only  half  as  much  work  as  a  man, 
how  many  hours  a  day  must  42  boys  work  to  accomplish 
as  much  in  45  dy.  as  27  men,  working  10  hr.  a  day,  would 
accomplish  in  28  dy.? 


RATIO   AND   PROPORTION.  229 


Proportional  Parts. 

399.  If  it  is  required  to  divide  a  quantity  into  parts 
proportional  to  3,  4,  5 ;  the  numbers  3,  4,  5  may  be  taken 
to  represent  the  parts,  and  then  the  whole  will  be  repre- 
sented by  3  +  4  +  5  ;   that  is,  by  12. 

400.  Examples.  1.  Divide  $391  into  parts  propor- 
tional to  the  numbers  5,  7,  and  11. 

Solution.    The  whole  quantity  will  be  represented  by  5 + 7  + 1 1  =23. 
Therefore,  the  respective  parts  will  be  £Jt  j/g,  |£  of  $391 ;  that  is, 
$85,  $119,  and  $187. 

Or,  the  parts  of  the  quantity  may  be  found  by  the  proportions : 

23:    5::  $391:? 

23:    7::  $391:? 

23:  11::  $391:? 
The  required  terms  will  be  $85,  $119,  and  $187,  respectively. 

2.    Divide  $248  into  parts  proportional  to  y1^,  y1^,  ^. 

Solution.  Multiply  the  fractions  by  150,  the  L.  C.  M.  of  their 
denominators.  The  results  are  15,  10,  6.  Hence;  the  parts  will  be 
represented  by  the  numbers  15,  10,  6,  and  the  whole  by  31. 

Therefore,  the  respective  parts  will  be  i^,  |f,  -fa  of  $248 ;  that  is, 
$120,  $80,  $48. 

Exercise  104. 

1.  Divide  $12,000  proportionally  to  the  numbers  3, 4,  5. 

2.  Divide  815  tons  proportionally  to  £,  §,  f ,  f . 

3.  Divide  6853  lb.  of  wool  proportionally  to  If,  2f ,  5| ; 
also  proportionally  to  the  reciprocals  of  these  numbers. 

4.  Two  men  purchase  some  property  together,  one  pay- 
ing $1250  and  the  other  $1000.  If  the  value  of  the  prop- 
erty rises  to  $3600,  what  will  be  the  share  of  each  ? 

5.  Gun  metal  is  composed  by  weight  of  3  parts  of  tin 
to  100  parts  of  copper.  What  weight  of  each  of  these 
metals  is  there  in  a  cannon  weighing  721  lb.? 


230  RATIO   AND   PROPORTION. 

6.  Bell  metal  contains  by  weight  78  parts  of  copper 
and  22  parts  of  tin.  What  weight  of  each  of  these  metals 
is  there  in  a  bell  weighing  937  lb.? 

7.  It  takes  75^  of  saltpetre,  12.5**  of  charcoal,  and 
12.5kg  of  sulphur  to  make  100**  of  powder.  .How  many 
kilograms  of  each  will  be  required  to  make  10,000,000 
cartridges,  each  containing  5*  of  powder  ? 

8.  Yellow  copper  contains  by  weight  2  parts  of  red 
copper  and  1  part  of  zinc.  How  many  ounces  of  red  cop- 
per in  an  article  of  yellow  copper  that  weighs  1  lb.? 

9.  Type  metal  is  an  alloy  containing  by  weight  39  parts 
of  lead  to  11  parts  of  antimony.  How  many  pounds  of 
each  are  required  to  make  957  lb.  of  type  ? 

10.  Plumbers'  solder  contains  by  weight  2  parts  of  lead 
and  1  part  of  tin.  How  many  pounds  of  each  are  required 
to  make  100  lb.  of  solder  ? 

11.  The  air  is  composed  of  oxygen  and  nitrogen.  In 
100  volumes  of  air  there  are  21  volumes  of  oxygen  and  79 
of  nitrogen.  If  the  weight  of  a  liter  of  oxygen  is  1.4295s, 
and  that  of  a  liter  of  nitrogen  is  1.2577g,  how  many  grams 
of  each  gas  does  100g  of  air  contain  ? 

12.  At  $20.67  an  ounce  for  pure  gold,  what  is  the  value 
of  the  gold  in  a  chain  that  weighs  3  oz.  4  dwt.,  if  it  is  18 
carats  fine  (that  is,  18  parts  of  pure  gold  out  of  24)  ? 

13.  Two  men  agree  to  do  a  piece  of  work  for  f  63.  They 
finish  the  work  in  18  days,  but  one  of  them  was  absent 
5  days  of  this  time.     How  should  the  pay  be  divided  ? 

14.  Five  men  working  together  do  a  piece  of  work  in 
20  days  and  receive  as  pay  $ 253.  One  of  the  men  was 
absent  5  days,  and  another  2  days  of  this  time.  How 
should  the  pay  be  divided  ? 

15.  Standard  silver  consists  of  37  parts  of  pure  silver 
to  3  parts  of  copper.  What  weight  of  pure  silver  in  the 
crown  piece  that  weighs  \<{  oz.  troy  ? 


RATIO   AND   PROPORTION.  231 

Partnership. 

401.  Partnership.  An  association  of  two  or  more  per- 
sons for  the  purpose  of  conducting  business  is  called  a 
partnership. 

A  partnership  association  is  called  a  firm,  company,  or 
house,  and  the  persons  associated  in  business  are  called 
partners. 

402.  Assets  and  Liabilities.  The  property  of  all  kinds 
of  a  firm  or  company  together  with  all  amounts  due  it  is 
called  its  assets  ;  the  debts  of  a  firm  or  company  are  called 
its  liabilities. 

403.  Partnership  is  separated  into  simple  and  compound. 
In  simple  partnership  the  capital   of   each  partner  is 

invented  for  the  same  time. 

In  compound  partnership  the  time  for  which  the  capital 
of  each  partner  is  invested  is  taken  into  account,  as  well  as 
the  amount  of  the  capital. 

404.  The  division  of  profits  and  losses  is  made  proportion- 
ally to  the  amount  of  the  capital  and  the  time  it  is  invested. 

405.  Examples.  1.  A  and  B  entered  into  partnership, 
A  furnishing  $4000,  and  B  $5000.  If  they  gained  $1800, 
what  was  each  partner's  share  of  the  profits  ? 

Solution.     $4000  +  $5000  =  $9000,  the  entire  capital. 
A's  share  of  the  profits  was  f£gg  of  $1800,  or  $800  ;   and  B's  share 
of  the  profits  was  g###  of  $1800,  or  $1000. 

2.  A  and  B  entered  into  partnership,  A  furnishing  $2000 
for  2  yr.,  and  B  $3000  for  1  yr.  If  their  profits  are  $1400, 
what  is  the  share  of  each  ? 

Solution.  The  use  of  $2000  for  2  yr.  is  equal  to  the  use  of 
2  X  $2000,  or  $4000,  for  1  yr.  Hence,  the  entire  capital  for  1  yr. 
equals  $4000  +  $3000,  or  $7000.  A's  share  of  the  profits  is  $$#  of 
$1400,  or  $800  ;   and  B's  share  is  f£#£  of  $1400,  or  $600. 


232  RATIO   AND   PROPORTION. 


Exercise  105. 

1.  A,  B,  and  C  entered  into  partnership,  A  furnishing 
$18,150  ;  B,  $19,360  ;  and  C,  $10,890.  If  their  profits 
were  $12,100,  what  was  each  man's  share  of  the  profits  ? 

2.  Four  men  engaged  in  business  together  and  made  a 
profit  of  $1200.  How  much  of  it  should  each  man  receive, 
if  the  first  put  in  $3000,  the  second  $5000,  the  third  $4200, 
and  the  fourth  $2400  ? 

3.  A  man  dies  owing  three  creditors  $8050,  $2970,  and 
$7170,  respectively.  If  his  assets,  after  deducting  ex- 
penses, are  $13,646,  how  much  will  each  creditor  receive? 

4.  Three  heirs  receive  from  an  estate  $4700,  $3200, 
and  $12,500,  respectively,  on  condition  that  they  together 
pay  a  debt  of  $2000.     What  amount  will  each  have  ? 

5.  Arnold  and  Baker  enter  into  partnership.  Arnold 
puts  in  $6000  for  8  mo.,  and  Baker  $4000  for  6  mo.  Their 
profits  are  $2000.     What  is  each  man's  share  ? 

6.  Dobson  furnishes  the  firm  of  Dobson  &  Fogg  with 
$5000  for  13  mo.;  Fogg  furnishes  $7000  for  9  mo.  Their 
profits  are  $1700.     What  is  the  share  of  each  ? 

7.  In  a  business  partnership,  A  furnishes  $800,  and 
after  3  mo.  $250  more  ;  B  furnishes  $950,  and  at  the  end 
of  2  mo.  withdraws  $200 ;  C  furnishes  $650,  and  at  the 
end  of  6  mo.  $400  more.  At  the  end  of  the  year  their 
profit  is  $2516.     How  shall  it  be  divided  among  them  ? 

8.  Two  partners,  A  and  B,  enter  into  partnership  with 
capitals  of  $3500  and  $8700,  respectively,  and  A  is  to  have 
0.12  of  the  profits  for  managing  the  business.  How  shall 
a  profit  of  $1906.25  be  divided  between  them  ? 

9.  A  puts  $2100  into  a  business,  and  B  $1750.  At 
the  end  of  a  year  each  puts  in  $700  more,  and  C  joins 
them  with  $2500.  How  shall  a  profit  of  $2166.50  be 
divided  18  months  after  C  enters  the  firm  ? 


RATIO   AND   PROPORTION.  233 

10.  Three  graziers  hire  a  pasture,  for  which  they  pay 
$132.50.  One  puts  in  10  oxen  for  3  months,  another  12 
oxen  for  4  months,  and  the  third  14  oxen  for  2  months. 
How  much  of  the  rent  ought  each  to  pay  ? 

11.  A  begins  business,  with  a  capital  of  $2400,  on  the 
19th  of  March;  and  on  the  17th  of  July  admits  B  as  a 
partner,  with  a  capital  of  $1800.  December  31  the  profits 
are  $943.     What  is  t^ie  share  of  each  ? 

12.  A  and  B  join  capitals  in  the  ratio  7:11.  At  the 
end  of  7  months  A  withdraws  \  of  his,  and  B  £  of  his  ; 
and,  after  11  months  more,  they  divide  a  profit  of  $5148.50. 
What  is  the  share  of  each  ? 

13.  Divide  £65  9s.  among  three  men,  so  that  the  first 
may  have  as  many  half-crowns  as  the  second  has  shillings  ; 
and  the  second  as  many  guineas  as  the  third  has  pounds. 

14-.  A  and  B  begin  business  each  with  a  capital  of  $2000. 
A  adds  $500  at  the  end  of  2  months,  and  $500  more  at  the 
end  of  7  months  ;  B  adds  $800  at  the  end  of  3  months. 
If  the  profits  are  $3605.25  at  the  end  of  a  year,  what  is 
the  share  of  each  ? 

15.  Three  partners  in  a  restaurant  furnish  respectively 
$500  for  7  months,  $600  for  8  months,  and  $900  for  9 
months.  If  they  lose  $410,  what  is  each  one's  share  of 
the  loss  ? 

16.  Two  capitalists  contribute,  one  $10,000,  the  other 
$12,000,  to  an  enterprise  which  continues  in  operation  for 
10  years.  10  months  after  starting,  a  third  man  becomes  a 
partner  and  contributes  $15,000;  and  2  years  after  this  a 
fourth  man  contributes  $17,400.  If  the  total  profits  are 
$45,600,  what  amount  does  each  partner  receive  ? 

17.  A  began  business  with  a  capital  of  $2500.  After 
three  years  he  invested  $1250  more,  and  took  as  a  partner 
B,  who  invested  $5000.  At  the  end  of  four  years  more  the 
profits  amounted  to  $9562.50.   What  was  the  share  of  each? 


234  RATIO   AND   PROPORTION. 


Averages,  or  Alligation. 

406.  The  average  of  several  numbers  is  the  number 
that  can  be  put  in  place  of  each  of  them  without  altering 
their  sum. 

Thus,  the  average  weight  of  four  turkeys,  weighing  respectively 

10  lb.,  11  lb.,  12  lb.,  and  13  lb.,  is  11|  lb.;  for  the  four  turkeys 
together  weigh  10  lb.  +  11  lb.  +  12  lb.  +  13  lb.,  or  46  lb.,  and  the 
average  weight  of  the  four  turkeys  is  ±  of  46  lb.,  or  11$  lb. 

407.  Alligation  is  the  process  of  finding  the  average 
value  of  a  compound  or  mixture  composed  of  quantities  of 
different  values;  or  of  finding  the  proportion  of  several 
quantities  of  different  values  that  must  be  used  to  form  a 
compound  or  mixture  of  a  given  average  value. 

Note.  The  first  process  is  called  alligation  medial;  the  second 
process,  alligation  alternate. 

408.  Examples.  1.  A  grocer  mixed  11  lb.  of  coffee, 
costing  $0.25  a  pound,  with  6  lb.  of  chicory,  costing  $0.08 
a  pound.      What  was  the  cost  of  the  mixture  a  pound  ? 

11  X  $0.25  =  $2.76  Solution.     11  lb.  at  $0.26  cost  $2.76,  and 
6  X    0.08  =    0.48        6  lb.  at  $0.08  cost  $0.48.     Therefore,  17  lb.  of 

17  X     $  ?  =  $3.23        the  mixture  cost  $3.23,  or  $0.19  a  pound. 

2.  In  what  proportion  may  a  grocer  mix  syrups  costing 
respectively  42,  56,  65,  and  75  cents  a  gallon  to  make  a 
mixture  worth  60  cents  a  gallon  ? 

Solution.  1  gal.  of  the  42-cent  syrup  gains  in  value  $0.18,  and 
1  gal.  of  the  75-cent  syrup  loses  in  value  $0. 15.  Hence,  to  make  a  mix- 
ture of  these  two  syrups  worth  $0.60  a  gallon,  the  grocer  must  mix 
them  in  the  ratio  15  :  18 ;  that  is,  6  :  6. 

1  gal.  of  the  66-cent  syrup  gains  in  value  $0.04,  and  1  gal.  of  the 
65-cent  syrup  loses  in  value  $0.05.  Hence,  to  make  a  mixture  of  these 
two  syrups  worth  $0.60  a  gallon,  he  must  mix  them  in  the  ratio  5:4. 

Therefore,  the  grocer  may  take  5  gal.  at  $0.42,  5  gal.  at  $0.66, 
4  gal.  at  $0.65,  and  6  gal.  at  $0.75. 


RATIO   AND   PROPORTION.  235 


Exercise  106. 


1.  There  were  125  pupils  at  school  on  Monday,  130  on 
Tuesday,  128  on  Wednesday,  132  on  Thursday,  and  125  on 
Friday.     What  was  the  average  daily  attendance  ? 

2.  A  spring  of  water  that  yields  250  gal.  an  hour 
supplies  a  town  containing  360  families.  What  is  the 
average  daily  supply  of  water  for  each  family  ? 

3.  A  wine  merchant  put  into  an  empty  cask  15  qt.  of 
brandy  costing  $1.10  a  quart,  66  qt.  costing  $1.20  a  quart, 
and  43  qt.  costing  $  1.40  a  quart.  At  what  price  per  quart 
must  he  sell  the  brandy  to  gain  one  fifth  of  the  cost  ? 

4.  A  grocer  mixed  120  lb.  of  tea  costing  50  cents  a 
pound  with  180  lb.  costing  40  cents  a  pound.  At  what 
price  per  pound  must  he  sell  the  mixture  to  make  a  profit 
of  $30  on  the  whole  ? 

5.  A  grocer  buys  two  kinds  of  tea  at  40  cents  a  pound 
and  56  cents  a  pound,  respectively,  and  mixes  them  in  the 
ratio  of  5  to  3.  What  is  his  profit,  if  he  sells  56  lb.  of  the 
mixture  at  84  cents  a  pound  ? 

6.  The  average  length  of  ten  sticks  is  2  ft.  10£  in.;  one 
stick  is  27£  in.  long,  another  37-J  in.  long,  and  the  remain- 
ing eight  are  of  the  same  length.  What  is  the  length  of 
one  of  the  remaining  eight? 

7.  The  average  age  of  the  boys  in  the  four  classes  of 
a  school  is  18.4  yr.,  17.9  yr.,  16.8  yr.,  and  15.7  yr.  The 
classes  contain  29,  33,  34,  and  33  boys,  respectively.  What 
is  the  average  age  of  the  boys  in  the  school  ? 

8.  Seven  boys  weigh  respectively  119.7  lb.,  105  lb., 
178.3  lb.,  165.3  lb.,  142.8  lb.,  109  lb.,  154.2  lb.  What  is 
their  average  weight  ? 

9.  In  what  proportion  should  tea  costing  60  cents  a 
pound  be  mixed  with  tea  costing  45  cents  a  pound  that 
the  cost  of  the  mixture  should  be  54  cents  a  pound  ? 


236  RATIO   AND   PROPORTION. 

10.  A  merchant  has  teas  that  cost  80  cents,  60  cents, 
and  40  cents  a  pound,  respectively.  How  many  pounds  of 
each  kind  shall  he  take  to  make  a  mixture  of  1000  lb.,  so 
that  in  selling  it  at  70  cents  a  pound  he  may  make  a  profit 
of  8  cents  a  pound  ? 

11.  A  grocer  mixed  black  tea  that  cost  him  28  cents  a 
pound  with  green  tea  that  cost  him  42  cents,  and  by  sell- 
ing the  mixture  at  35  cents  a  pound  he  gained  £  of  its 
cost.  What  was  the  actual  cost  of  the  mixture  a  pound  ? 
In  what  ratio  were  the  teas  mixed  ? 

12.  A  dealer  has  an  order  for  1000  bu.  of  wheat  at  70 
cents  a  bushel.  In  what  proportion  shall  he  mix  three 
kinds  of  wheat  at  66,  69,  and  72  cents  a  bushel  to  fill  the 
order  ? 

13.  A  wine  merchant  mixes  wines  that  cost  $0.95,  $1.05, 
$1.10,  and  $1.20  a  gallon  to  make  a  mixture  costing  $1.00 
per  gallon.  How  many  gallons  of  each  kind  of  wine  does 
he  take  ? 

14.  A  merchant  wishes  to  fill  a  barrel  that  will  hold 
240  lb.  of  sugar  with  sugar  costing  4 J,  4£,  and  5 £  cents 
a  pound,  respectively,  so  that  the  mixture  may  cost  4$-  cents 
a  pound.     How  many  pounds  of  each  kind  shall  he  take? 

15.  A  grocer  wishes  to  mix  12  lb.  of  coffee  at  40  cents  a 
pound  and  20  lb.  at  35  cents  a  pound  with  coifee  at  28 
cents  a  pound,  so  that  the  mixture  may  be  worth  30  cents 
a  pound.     How  many  pounds  at  28  cents  must  he  use  ? 

16.  A  grocer  mixed  14  lb.  of  coffee  costing  32  cents  a 
pound,  18  lb.  costing  35  cents  a  pound,  22  lb.  costing  38 
cents  a  pound,  and  40  lb.  costing  30  cents  a  pound.  What 
is  the  cost  of  the  mixture  per  pound,  and  at  what  price 
must  he  sell  it  to  gain  0.25  of  the  cost  ? 

17.  In  what  proportion  may  oils  costing  $1.20,  $0.80, 
and  $0.60  a  gallon  be  mixed  that  the  mixture  may  cost 
$0.70  a  gallon  ? 


CHAPTER   XII. 
PERCENTAGE. 

409.  A  percentage  of  a  number  is  the  result  obtained 
by  taking  a  stated  number  of  hundredths  of  it. 

One  hundredth  of  a  number  is  called  one  per  cent  of  it ; 
two  hundredths,  two  per  cent ;   and  so  on. 

410.  A  rate  per  cent  is  a  fraction  whose  denominator 
is  100,  and  whose  numerator  is  the  given  number  of  hu?i- 
dredths.  The  methods  of  common  fractions  or  of  decimals 
are  used  in  the  solution  of  all  examples  in  Percentage. 

The  shortest  method  is  the  best  method. 

411.  The  symbol  %  stands  for  the  words  per  cent. 
Thus,  13%  is  0.13  ;   2£%  is  0.02i;   867%  is  8.67. 

412.  Example.     Express  37-J-%  as  a  common  fraction. 

37-i-      q 
Solution.  37£%  =  —J  =  '-  •    Hence, 

413.  To  Express  a  Rate  Per  Cent  as  a  Common  Fraction, 

Write  the  rate  for  the  numerator  and  100  for  the  denomi- 
nator, and  reduce  this  fraction  to  its  lowest  terms. 

Exercise  107. 
Reduce  to  a  common  fraction  : 


1. 

20%. 

6. 

5%. 

11. 

62^%. 

16. 

18f%. 

2. 

80%. 

7. 

10%. 

12. 

87^%. 

17. 

95%. 

3. 

25%. 

8. 

12*%. 

13. 

66f%. 

18. 

70%. 

4. 

50%. 

9. 

16f%. 

14. 

37^%. 

19. 

144|% 

5. 

75%. 

10. 

114%. 

15. 

83£%. 

20. 

262£% 

238 


PERCENTAGE. 


414.    Example.     Express  £  as  a  rate  per  cent. 


Solution. 


i  =  m  =  100%. 

|  =  |  of  100%  =  40%.     Hence, 


415.  To  Express  a  Common  Fraction  as  a  Rate  Per 
Cent, 

Divide  100  by  the  denominator  of  the  fraction  and  mul- 
tiply the  quotient  by  the  numerator. 

416.  Examples.     1.   Express  0.4  as  a  rate  per  cent. 
Solution.     0.4  =  0.40,  or  40%. 

2.  Express  0.4575  as  a  rate  per  cent. 
Solution.     0.4575  =  0.45^0  =  0.45|,  or  45|%. 

3.  Express  0.00375  as  a  rate  per  cent. 
Solution.     0.00375  =  O.OO&Vff  =  O.OOf  =  |%.     Hence, 

417.  To  Express  a  Decimal  as  a  Rate  Per  Cent, 
Write  the  decimal  as  hundredths,  and  the  number  that 

exj>resses  the  hundredths  is  the  rate  per  cent  required. 

Note.     If  the  decimal  has  more  than  two  places,  the  figures  that 
follow  the  hundredths'  place  signify  a  fraction  of  1%. 


Exercise  108. 

lX[ 

res£ 

as  a  rate  per 

cent : 

1. 

i- 

8-    t- 

15.    0.25. 

22. 

0.33333 

2. 

i- 

9-    A- 

16.    0.6. 

23. 

0.16667 

3. 

i- 

io.  a- 

17.    0.75. 

24. 

0.83333 

4. 

i- 

11.  1. 

18.    0.9. 

25. 

0.875. 

5. 

i 

12.    TV 

19.    0.65. 

26. 

1.375. 

6. 

f 

13.     T*p 

20.    0.45. 

27. 

2.66667. 

7. 

§• 

14.   a- 

21.    0.2. 

28. 

4.2525. 

PERCENTAGE.  239 

418.  Problems  in  Percentage  are  conveniently  divided 
into  three  classes,  as  follows  : 

Class      I.  To  find  a  certain  fraction  of  a  number. 

Class  II.  To  find  the  fraction  that  one  number  is  of 
another. 

Class  III.  To  find  a  number  when  a  fraction  of  it  is  given. 

The  following  examples  illustrate  the  three  classes  : 

Class      I.    What  number  is  f  of  300  ? 

Class    II.    What  fraction  of  300  is  200  ? 

Class  III.   What  is  the  number,  if  200  is  |  of  it  ? 

The  fraction  in  each  class  is  expressed  in  hundredths  for 
the  sake  of  a  uniform  standard  ;  the  phrase  per  cent  is  used 
for  tjie  word  hundredths,  and  the  symbol  °J0  is  written  for 
the  phrase  per  cent. 

Thus,  in  common  fractions  8  is  £  of  16 ;  in  decimals  &  is  0.5  of  16 ; 
in  percentage  8  is  50%  of  16. 

In  common  fractions  5  is  £  of  40  ;  in  decimals  5  is  0.125  of  40  ;  in 
percentage  5  is  12|%  of  40. 

419.  One  hundred  per  cent  of  a  number  is  the  number 
itself. 

Thus,  100%  of  40  is  40. 

Class  I. 

420.  Examples. 


1.    Findl6i%  of  288. 

Solution.     16£%  =  0.16f 

0.16i  of  288  =  47.04. 
Therefore,  16£%  of  288  is  47.04. 


2.    Find  16J%  of  288. 

Solution.     16f-%  =  0. 16$-  *  \. 

\  of  288  =  48. 
Therefore,  16£%  of  288  is  48. 


421.   To  Find  a  Percentage  of  a  Number, 

Multiply  the  number  by  the  given  rate  per  cent,  expressed 
as  a  common  fraction  or  as  a  decimal. 


240  PERCENTAGE. 

Exercise  109. 
Find  by  using  decimals  : 

1.  23%  of  1728.  6.  2%  of  846.  11.  0.5%ofl44. 

2.  44%  of  1861.  7.  9%  of  24.87.  12.  8752%  of  2645. 

3.  87%  of  14.22.  8.  122%  of  12.5.  13.  0.02%  of  52.36. 

4.  63%  of  2.832.  9.  287%  of  48.2.  14.  2%  of  3. 

5.  72%  of  841.  10.  1%  of  7854.  15.  2.06%  of  312. 

Find  by  using  common  fractions  : 

16.  33£%of363.  21.  62i%of216.  26.  14f%of81.9. 

17.  20%  of  545.  22.  37£%of360.  27.  22§%of8.19. 

18.  25%  of  1728.  23.  83£%of486.  28.  168^%  of  256. 

19.  50%  of  8642.  24.  66f  %  of  456.  29.  143|  %  of  288. 

20.  75%  of  432.  25.  12^%  of  2.56.  30.  70%  of  8432. 

31.  The  population  of  a  town  in  1880  was  12,275,  and 
it  increased  8%  in  the  next  ten  years.  Find  the  popula- 
tion of  the  town  in  1890. 

32.  How  much  metal  will  be  obtained  from  365  tons  of 
ore,  if  the  ore  contains  7%  of  metal? 

33.  If  gunpowder  contains  75%  of  saltpetre,  10%  of 
sulphur,  15%  of  charcoal,  how  many  pounds  of  each  are 
there  in  a  ton  of  powder  ? 

34.  Air  is  composed  by  volume  of  20.0265%  of  oxygen 
and  79.9735%  of  nitrogen.  How  many  cubic  feet  of  oxy- 
gen in  1750  cu.  ft.  of  air  ? 

85.  If  2%  of  a  regiment  of  750  men  are  killed  in  an 
engagement,  6%  are  wounded,  and  4%  are  missing,  what 
is  the  number  still  available  for  service  ? 

36.  A  man  sold  a  bicycle  that  cost  him  $60,  and  lost 
16f%  0I"  the  cost.     For  what  price  did  he  sell  it  ? 

37.  A  merchant  sold  hats  that  cost  him  $1.50  each,  and 
gained  33£%.     For  what  price  did  he  sell  them  ? 


PERCENTAGE.  241 

38.  In  a  school  of  80  children,  17£%  are  girls.  Find 
the  number  of  boys. 

39.  The  lead  ore  from  a  certain  mine  yields  60%  of 
metal,  and  of  the  metal  f  of  1%  is  silver.  How  much 
silver  and  how  much  lead  will  be  obtained  from  1200  t. 
of  ore  ? 

40.  If  13%  of  a  population  of  27,000,000  are  foreign 
born,  how  many  of  the  population  are  foreign  born  ? 

41.  If  iron  expands  J  of  l°/0  when  heated  185°  F.,  what 
will  be  the  expansion  of  iron  when  heated  from  —  20°  F. 
to+120°F.? 

42.  A  tubular  iron  bridge  740  ft.  long  has  one  end  fast 
to  a  pier.  How  much  play  must  be  allowed  at  the  other 
end  for  the  expansion  of  the  iron,  if  the  climate  varies 
from  —  30°  F.  in  winter  to  +  130°  F.  in  a  July  sun  ? 

43.  How  much  longer  is  100  miles  of  iron  rails  at  118° 
F.  than  at  20°  below  zero  ? 

Class  II. 

422.  Examples.     1.   What  per  cent  of  4  is  3  ? 

Solution.     3  is  f  of  4,  and  £  =  T7^,  or  75%. 
Therefore,  3  is  75%  of  4. 

2.   What  per  cent  of  447  is  169.86  ? 

Solution.      4.47  =  1%  of  447. 

Therefore,  169.86  =  ^Tgr  %  of  447  =  38%  of  447.     Hence, 

423.  To  Find  the  Rate  Per  Cent  when  a  Number  and 
a  Percentage  of  the  Number  are  Given, 

Write  the  number  as  the  denominator  and  the  percentage 
as  the  numerator  of  a  fraction,  and  reduce  this  fraction  to 
hundredths.     Or, 

Divide  the  percentage  of  the  number  by  one  per  cent  of  the 
number. 


hTf 


242  PERCENTAGE. 


Exercise  110. 

1.  What  per  cent  of  64  is  16  ? 

2.  What  per  cent  of  16  is  64  ? 

3.  What  per  cent  of  450  lb.  is  50  lb.? 

4.  What  per  cent  of  50  lb.  is  450  lb.? 

5.  What  per  cent  of  $465  is  $130.20  ? 

6.  What  per  cent  of  $832  is  $807.04  ? 

7.  What  per  cent  of  $987  is  $2289.84  ? 

8.  A  brick  kiln  contained  29,800  bricks,  but  after  burn- 
ing only  29,734  were  found  in  good  condition.  What  per 
cent  had  been  spoiled  in  burning  ? 

9.  If  a  house  worth  $4000  rents  for  $360  a  year,  what 
per  cent  of  its  value  is  the  rent  ? 

10.  If  75  bu.  of  corn  are  raised  from  1  pk.  of  corn,  what 
per  cent  is  the  increase  ? 

11.  Ten  years  ago  the  population  of  a  city  was  26,275  ; 
its  present  population  is  31,530.  What  is  the  increase  per 
cent  ? 

12.  If  3f  tons  of  sulphur  are  required  to  make  31  J-  tons 
of  gunpowder,  what  per  cent  of  gunpowder  is  sulphur  ? 

13.  If  a  long  ton  of  ore  in  a  gold  mine  yields  5  oz.  (troy) 
of  gold,  what  is  the  yield  per  cent  ? 

14.  If  12£  tons  of  iron  are  obtained  from  235  tons  of 
ore,  what  per  cent  of  the  ore  is  iron  ? 

Find  the  gain  per  cent  in  population  in  each  of  the  fol- 
lowing cities  from  1880  to  1890  : 


Cities. 

1880. 

1890. 

15. 

New  York, 

1,206,594, 

1,513,501. 

16. 

Chicago, 

503,304, 

1,099,850. 

17. 

Philadelphia, 

846,981, 

1,046,964. 

18. 

Brooklyn, 

566,689, 

806,343. 

19. 

Boston, 

362,535, 

448,477. 

PERCENTAGE.  243 

20.  If  2  gal.  of  water  are  added  to  25  gal.  of  alcohol, 
what  per  cent  of  the  mixture  is  water  ?  What  per  cent  is 
alcohol  ? 

21.  If  5%  of  the  present  population  of  a  town  has  been 
the  increase  in  the  preceding  ten  years,  what  per  cent  of 
the  population  ten  years  ago  has  been  added  ? 

22.  A  man  gained  in  weight  in  January  3%,  and  in 
February  lost  3%.  What  per  cent  of  his  weight  the  first 
day  of  January  is  his  weight  the  first  day  of  March  ? 

23.  If  7  lb.  of  a  certain  article  loses  3  oz.  in  weight  by 
drying,  what  per  cent  of  its  original  weight  is  water  ? 

24.  If  7  lb.  of  a  dry  article  has  lost  3  oz.  by  drying,  what 
per  cent  of  its  original  weight  was  water  ? 

25.  If  a  dry  article  exposed  to  damp  air  absorbed  3  ozt 
of  water,  and  then  weighed  7  lb.,  what  per  cent  of  its 
present  weight  is  water  ? 

26.  If  rosin  is  melted  with  20%  of  its  weight  of  tallow, 
what  per  cent  of  tallow  does  the  mixture  contain  ? 

27.  If  20%  of  a  mixture  of  tallow  and  rosin  is  tallow, 
what  per  cent  of  the  weight  of  the  rosin  is  the  weight  of 
the  tallow  ? 

28.  Nitrogen  gas,  under  standard  pressure  and  tempera- 
ture, is  -J-  of  1%  of  the  weight  of  an  equal  volume  of  water. 
What  is  the  specific  gravity  of  nitrogen  ?  How  many  gal- 
lons of  nitrogen  will  it  take  to  weigh  as  much  as  a  pint  of 
water  ? 

29.  Oxygen  gas  is  \  of  l°/0  of  the  weight  of  an  equal 
volume  of  water.  What  is  its  specific  gravity  ?  How 
many  gallons  of  oxygen  will  it  take  to  weigh,  as  much  as  a 
pint  of  water  ? 

30.  If  common  air  consists  of  4  volumes  of  oxygen  to 
13  of  nitrogen,  what  is  its  specific  gravity  ? 

31.  How  many  gallons  of  air  will  it  take  to  weigh  as 
much  as  a  pint  of  water  ? 


244  PERCENTAGE. 


Class  IIL 

424.  Examples.  1.  If  17%  of  a  number  is  799,  what 
is  the  number  ? 

Solution.  Since  17%  of  a  number  is  799,  1%  of  the  number  is  TJ7 
of  799,  or  47,  and  100%  of  the  number  is  100  X  47,  or  4700. 

2.  If  16f  %  of  a  number  is  432,  what  is  the  number  ? 

Solution.     432  is  16 1%,  or  £  of  the  required  number. 
Therefore,  the  required  number  is  6  X  432,  or  2592. 

3.  1400  is  16J%  more  than  what  number  ? 

Solution.     100  %  of  the  number  =  the  number. 

16%%  of  the  number =  the  increase. 

116|%,  or  I  of  the  number  =  1400. 

I  of  the  number  =  \  of  1400,  or  200. 
Therefore,  the  number  =  6  X  200,  or  1200. 

4.  1200  is  25%  less  than  what  number? 

Solution.     100%  of  the  number  =  the  number. 

25%  of  the  number =  the  decrease. 

75%,  or  £  of  the  number  =  1200. 
If  1200  is  £  of  the  number,  £  of  the  number  is  \  of  1200,  or  400. 
Therefore,  the  number  is  4  X  400,  or  1600.     Hence, 

425.  To  Find  a  Number  when  a  Percentage  of  the 
Number  and  the  Rate  Per  Cent  are  Given, 

Express  the  rate  per  cent  as  a  fraction,  divide  the  per- 
centage by  the  numerator  of  this  fraction  and  multiply  the 
quotient  by  the  denominator. 

Exercise  111. 

1.  15   is  f  of  what  number?     15  is  75%    of   what 
number  ? 

2.  $500  is  4%  of  what  sum  of  money  ? 

3.  Find  the  number  of  which  324  is  27%. 


PERCENTAGE.  245 

4.  288  is  20%  more  than  what  number  ? 

5.  145  is  25%  more  than  what  number  ? 

6.  1240  is  55%  less  than  what  number  ? 

7.  260  is  33-J%  less  than  what  number  ? 

8.  91  is  40%  more  than  what  number  ? 

9.  901  is  6£%  more  than  what  number  ? 

1 0.  If  8f  %  of  a  number  is  4140.15,  what  is  the  number  ? 

11.  If  3%  of  a  number  is  2f,  what  is  the  number  ? 

12.  If  140%  of  a  number  is  630,  what  is  the  number  ? 

13.  If  6£%  of  a  number  is  33.25,  what  is  the  number? 

14.  A  town,  after  decreasing  11%,  has  4539  inhabitants. 
Find  its  number  at  first. 

15.  In  a  certain  school  there  are  200  girls,  and  the 
number  of  girls  is  40%  of  the  whole  number  of  pupils. 
How  many  pupils  in  the  school  ? 

16.  A  manufactory  uses  24  tons  of  coal  a  day,  20%  of 
which  is  lost  in  smoke.  How  much  coal  would  be  needed 
if  this  waste  could  be  prevented  ? 

17.  A  town,  after  decreasing  25%,  has  4539  inhabitants. 
Find  its  number  at  first. 

18.  If  the  ore  from  a  mine  yields  -fa  of  1%  of  pure  gold, 
how  many  long  tons  of  ore  must  be  taken  to  obtain  7  lb. 
(troy)  of  gold  ? 

19.^  Goods  were  sold,  at  a  loss  of  3%,  for  $2667.50. 
What  was  the  cost  ? 

20.  A  tradesman,  in  selling  goods,  deducts  from  the 
marked  price  5%  for  cash.  What  was  the  marked  price 
of  goods  for  which  he  received  $14.25  ? 

21.  If  an  ore  loses  41^-%  of  its  weight  in  roasting,  and 
43f  %  of  the  remainder  in  smelting,  how  much  ore  will  be 
required  to  yield  1000  tons  of  metal  ? 

22.  How  many  pounds  of  tallow  must  be  mixed  with 
8 J  pounds  of  rosin  that  the  mixture  may  contain  15%  of 
tallow  ? 


246  PERCENTAGE. 

Commercial  Discount. 

426.  Commercial  Discount.  A  reduction  from  the  list 
price  of  an  article,  from  the  amount  of  a  bill  of  goods,  or 
from  the  amount  of  a  debt  is  called  commercial  discount. 

427.  Discounts  are  reckoned  at  some  common  fraction 
of,  or  at  some  rate  per  cent  of,  the  amount  from  which  the 
discount  is  made. 

If  two  or  more  discounts  are  quoted,  the  first  denotes  a 
discount  off  the  list  price ;  the  second,  a  discount  off  the 
remainder  after  the  first  discount  is  made  ;  the  third,  off  the 
remainder  after  the  second  discount  is  made  ;   and  so  on. 

Thus,  discounts  of  20  and  i  mean  that  20%  is  to  be  deducted  from 
the  amount,  and  then  from  the  remainder  ^  of  it  is  to  be  taken. 

Note.  By  varying  the  rate  of  discount  the  manufacturer  can  raise 
or  lower  the  price  of  his  goods  without  issuing  a  new  catalogue. 

428.  Examples.  1.  Find  the  net  amount  of  a  bill  of 
$150  after  a  discount  of  33£%  is  made. 

Solution.     Discount  is  33^%,  or  i  of  $160  =  $30. 
Net  amount  is  $160  —  $30         =  $120. 

2.  Find  the  net  amount  of  a  bill  of  $527.10  with  £  and 
20%  off. 

$527.10 

175.70  Solution.      $527.10  less  £  of  $527.10  is  $351.40. 

$351.40  20%  =  £,  and  $361.40  less  £  of  $351.40  is  $281.12,  the 

70.28  net  amount  of  the  bill. 


$281.12 


Exercise  112. 


1.  Find  the  net  amount  of  a  bill  of  $1550,  if  a  dis- 
count of  5%  is  made  for  cash. 

2.  Find  the  net  amount  of  a  bill  of  $88,  if  the  dis- 
counts are  20  and  10. 


PERCENTAGE.  247 

3.  Find  the  net  cash  amount  of  a  bill  of  $ 800,  if  the 
discounts  are  75,  5,  and  2\. 

4.  Find  the  net  cash  amount  of  a  bill  of  $272,  if  the 
discounts  are  -J-,  10,  and  5. 

5.  Find  the  net  cash  amount  of  a  bill  of  $1440,  if  the 
discounts  are  55,  10,  and  5. 

6.  Find  the  net  cash  amount  of  a  bill  of  $1125,  if  the 
discounts  are  -J,  10,  10,  10,  and  5. 

7.  Find  the  net  amount  of  a  bill  of  $872.29,  if  the 
discounts  are  -J-,  20,  and  25. 

8.  Find  the  difference  between  a  single  discount  of 
50%  and  two  successive  discounts  of  25°f0  an(i  25%  off  a 
bill  of  $1272.36. 

9.  An  agent  bought  25  sewing  machines  with  15,  10, 
and  5  off  the  list  price  of  $40  each,  and  sold  them  at  a 
discount  of  10%  off  the  list  price.  What  was  the  net 
amount  he  received  for  the  sewing  machines  and  his 
profit  ? 

10.  An  agent  bought  a  bicycle  with  25  and  5  off  the  list 
price  of  $100.  If  he  received  an  additional  discount  of 
2J%  for  cash,  and  sold  the  bicycle  at  a  discount  of  12-j-% 
off  the  list  price,  what  was  the  selling  price  and  his  profit  ? 

11.  A  collector  collects  6b %  of  a  debt  of  $727,  and 
charges  5%  of  the  amount  he  collected.  What  was  the 
net  amount  for  the  creditor  ? 

Gain  and  Loss. 

429.  The  gain  or  loss  in  business  transactions  is  often 
computed  as  a  per  cent  of  the  cost. 

430.  In  commercial  discount  the  per  cent  is  always 
reckoned  upon  the  price  asked.  In  gain  or  loss  the  per  cent 
is  always  reckoned  upon  the  price  paid  ;  that  is,  upon  the 
cost. 


248  PERCENTAGE. 


Exercise  113. 


1.  If  goods  are  bought  for  $415,  and  sold  for  $500, 
what  is  the  gain  per  cent  ? 

2.  If  goods  are  bought  for  $415,  and  sold  for  $400, 
what  is  the  loss  per  cent  ? 

3.  A  farmer  buys  24  head  of  cattle  at  $80  a  head. 
After  losing  6  head,  he  sells  the  remainder  at  $105  a  head. 
What  does  he  gain  or  lose  per  cent  ? 

4.  Teas  at  68  cents,  86  cents,  and  96  cents  a  pound  are 
mixed  in  equal  quantities,  and  sold  at  90  cents  a  pound. 
Find  the  gain  per  cent. 

5.  By  selling  goods  for  $1173.92  a  merchant  gains 
$153.12.     Find  the  gain  per  cent. 

6.  What  was  the  cost,  when  17£%  was  gained  by  sell- 
ing goods  for  $253.80  ? 

7.  A  wine  merchant  mixes  24  gal.  of  wine  at  $7  a  gal- 
lon, with  18  gal.  at  $5  a  gallon,  and  sells  the  whole  at  $7 
a  gallon.     What  does  he  gain  per  cent  ? 

8.  By  selling  a  horse  for  $200,  a  dealer  loses  12£%. 
What  would  he  have  gained  or  lost  per  cent  if  he  had  sold 
the  horse  for  $250  ? 

9.  A  spirit  merchant  buys  75  gal.  of  spirits  at  $3.25  a 
gallon,  and,  after  drawing  off  10  gal.,  sells  the  remainder 
so  as  to  gain  5%  on  the  cost  of  the  whole.  What  is  the 
selling  price  per  gallon  ? 

10.  A  man  owns  two  city  lots  worth  respectively  $9845 
and  $12,155.  If  the  first  gains  in  value  32%,  and  the  sec- 
ond loses  13%,  what  is  the  gain  or  loss  per  cent  in  the  value 
of  the  two  lots  ? 

11.  A  tradesman  marks  a  hat  $5,  but  takes  off  5%.  If 
his  profit  is  14%,  what  was  the  cost  of  the  hat? 

12.  What  would  a  dishonest  dealer  gain  per  cent  by 
using  a  false  weight  of  15  oz.  instead  of  a  pound  ? 


PERCENTAGE.  249 

13.  A  dishonest  dealer  gains  12%  by  using  false  weights. 
What  is  the  real  weight  of  his  pound  ? 

14.  What  per  cent  above  cost  must  a  merchant  mark 
his  goods  that  he  may  take  off  20%  from  the  marked  price, 
and  still  make  20%  on  the  cost  ? 

Solution.  Since  the  merchant  is  to  make  20%  on  the  cost  of  the 
goods,  the  selling  price  is  120%  of  the  cost  price. 

Since  the  selling  price  is  to  be  20%  below  the  marked  price,  the 
selling  price  is  80%  of  the  marked  price. 

Therefore,  the  marked  price  will  be  -^  of  120%  of  the  cost  price, 
or  150%  of  the  cost  price ;  that  is,  the  goods  must  be  marked  50% 
above  cost. 

15.  What  per  cent  above  cost  must  a  merchant  mark 
his  goods  to  take  off  10%,  and  still  gain  17%  ? 

16.  What  per  cent  above  cost  must  a  merchant  mark 
his  goods  to  take  off  12£%,  and  still  gain  12  J  %  ? 

17.  What  per  cent  above  cost  must  a  merchant  mark 
his  goods  to  take  off  15%,  and  still  gain  15%  ? 

18.  What  per  cent  above  cost  must  a  merchant  mark 
his  goods  to  take  off  33£%,  and  still  gain  33£%  ? 

19.  A  man  bought  a  horse  for  $70,  and  sold  him  for 
$80.  What  per  cent  did  he  gain  ?  What  per  cent  of  the 
selling  price  of  the  horse  did  he  gain  ? 

2Q.  If  a  merchant  clears  $800  by  selling  goods  for  12^-% 
profit,  what  was  the  cost  of  the  goods,  and  for  how  much 
were  they  sold  ? 

21.  A  man  selling  eggs  at  $0.40  a  dozen  gains  33J%  ; 
what  was  the  cost  ?  Another,  selling  at  the  same  price, 
gains  33^-%  of  his  receipts  ;    what  did  his  eggs  cost  ? 

22.  A  man  lost  10%  by  selling  a  carriage  for  $117. 
At  what  price  should  he  have  sold  it  to  make  10%  ? 

23.  If  a  real  estate  dealer  gained  $600  by  selling  a  farm 
for  20%  profit,  what  was  the  cost  of  the  farm,  and  for  how 
much  did  he  sell  it  ? 


250  PERCENTAGE. 


Commission. 

431.  Commission  is  the  payment  made  by  one  person, 
called  the  Principal,  to  another,  called  the  Agent  or  Factor, 
for  the  transaction  of  business. 

432.  Commission  is  usually  a  percentage  of  the  money 
involved  in  the  transaction.  If  goods  are  bought,  it  is  a 
percentage  of  the  amount  paid;  if  sold,  a  percentage  of  the 
amount  received;  if  money  is  collected,  a  percentage  of  the 
amount  collected. 

Exercise  114. 

1.  Find  the  commission  on  $2595,  at  2£%. 

2.  An  agent  sells  200  bbl.  of  flour,  at  $6.25,  and  600 
gal.  of  molasses,  at  65  cents,  and  charges  a  commission  of 
If  %.     What  are  the  net  proceeds  ? 

Note.  The  sum  left  after  the  payment  of  the  commission  and  of 
all  other  expenses  is  called  the  net  proceeds. 

3.  A  commission  merchant  received  $1640  to  buy  corn, 
and  charged  a  commission  of  2£%.  What  is  his  commis- 
sion, and  how  many  bushels  of  corn  at  62£  cents  a  bushel 
can  he  buy  ? 

4.  An  agent  sells  a  consignment  of  cotton  for  $5216. 
He  pays  $51  for  storage,  and  charges  a  commission  of  2£%. 
What  are  the  net  proceeds  ? 

5.  An  agent  sold  butter  for  $1570,  and  remitted 
$1546.45.     What  was  the  rate  per  cent  of  commission  ? 

6.  What  are  the  net  proceeds  from  the  sale  of  2250  bbl. 
of  flour  at  $6.25  a  barrel,  if  the  charge  for  freight  is  50 
cents  a  barrel,  the  commission  for  selling  2%,  and  the 
commission  for  guaranteeing  payment  1£%  ? 

7.  An  agent  sells  350  crates  of  peaches,  at  $2.60.  If 
the  commission  is  4-£-<j£,  find  the  net  proceeds. 


^ERCEtfTAGE.  251 

8.  An  agent  sells  420  acres  of  land,  at  $40  an  acre,  and 
charges  l\°f0  commission.     What  is  his  commission  ? 

9.  An  agent,  charging  k\°f0  commission,  receives  for 
his  services  $313.     Find  the  amonnt  of  his  sales. 

10.  A  merchant  buys  730  yd.  of  carpeting  at  $1.25  a 
yard,  and  pays  his  agent  f  of  l°/0  commission.  If  the 
freight  amounts  to  $23.58,  at  what  price  per  yard  must  he 
sell  the  carpeting  to  gain  20%  ? 

11.  An  agent  sells  a  consignment  of  goods  for  $2100. 
He  pays  $33.50  for  freight,  and  remits  $2024.50.  Find 
his  rate  of  commission. 

12.  An  agent  sells  5000  lb.  of  cotton  at  14  cents  a  pound, 
charging  2%  commission.  With  the  net  proceeds  he  buys 
cotton  cloth  at  10  cents  a  yard,  charging  \\°f0  commission. 
How  many  yards  of  cloth  does  he  buy  ? 

13.  An  agent  sold  500  bbl.  of  flour  at  $5.50  a  barrel,  and 
charged  2\°j0  commission  ;  the  expenses  for  freight,  etc., 
were  $250.  With  the  net  proceeds  he  bought  sugar  at  4f 
cents  a  pound,  charging  2-J-%  commission.  How  much 
sugar  did  he  buy,  and  what  was  his  total  commission  ? 

14.  A  collector's  commission  for  collecting  taxes,  at 
1£%,  is  $206.55.     What  sum  did  he  collect  ? 

15.  An  agent  received  $2961  to  purchase  goods,  and 
charged  5%  commission.     What  was  his  commission? 

l£.  An  agent  buys  3100  bbl.  of  flour  at  $4.50  a  barrel, 
and  charges  \\°j0  commission.     What  is  his  commission? 

17.  A  broker  receives  $6150  to  invest  in  cotton,  at  7f 
cents  a  pound.  If  his  commission  is  2£%,  how  many 
pounds  of  cotton  can  he  buy  ? 

18.  An  agent  sells  1100  bbl.  of  flour  at  $4.50  a  barrel, 
and  charges  2-j-%  commission.  He  invests  the  proceeds  in 
steel  at  \.\  cents  a  pound,  charging  l-j-%  commission.  What 
is  his  entire  commission,  and  how  many  long  tons  of  steel 
does  he  buy  ? 


252  PERCENTAGE. 

Insurance. 

433.  Insurance  is  the  guarantee  by  an  insurance  com- 
pany of  the  payment  of  a  stated  amount  to  the  person 
insured,  in  the  event  of  loss  of  property  by  fire,  by  storm 
at  sea,  or  by  other  specified  disaster ;  or  in  the  event  of 
the  death  of  the  person  insured,  or  of  an  accident  to  him. 

434.  The  policy  is  the  written  agreement. 

The  face  of  the  policy  is  the  sum  named  therein  which 
is  to  be  paid  to  the  holder  in  case  of  loss. 

435.  The  premium  is  the  sum  paid  for  the  insurance. 
It  is  reckoned  as  a  percentage  on  the  face  of  the  policy. 

436.  Insurance  companies  usually  insure  £  or  £  of  the 
value  of  the  property  ;  and,  in  case  of  partial  loss,  pay 
only  the  value  of  the  property  destroyed. 

Note.  Insurance  agents  are  also  called  underwriters,  and  insur- 
ing is  often  called  underwriting. 

Exercise  115. 

1.  Find  the  premium  of  the  fire  insurance  on  a  house 
for  $2650  at£of  1%. 

2.  Find  the  premium  for  insuring  a  man's  life  for 
$2500,  at  an  age  for  which  the  rate  is  2J%. 

3.  At  6£  %,  what  premium  will  be  paid  on  a  vessel 
worth  $36,400,  insured  for  £  its  value  ? 

4.  A  vessel  worth  $16,000  is  insured  for  £  its  value  at 
?i%.     What  is  the  premium  ? 

5.  The  premium  of  insurance  at  1£%  *s  $150.  What 
is  the  amount  insured  ? 

6.  A  vessel  valued  at  $128,000  is  insured  for  £  its 
value  at  3  J °f0.  What  is  the  net  loss  to  the  owners,  if  the 
vessel  is  destroyed  during  the  third  year  after  it  is  insured  ? 


PERCENTAGE.  253 

7.  A  building  worth  $7500  is  insured  at  f  its  value,  at 
■J  of  1%  per  annum.     What  is  the  annual  premium  ? 

8.  Four  companies  insure  a  store  and  contents  for 
$60,000.  One  company  takes  $20,000,  at  £  of  1%  ;  a 
second  takes  $10,000,  at  f  of  1%  ;  a  third,  $15,000,  at  { 
of  l°/0  ;  a  fourth,  the  remainder,  at  ^  of  1%.  What  is 
the  premium  ? 

9.  If  the  store  of  Ex.  8  is  damaged  to  the  extent  of 
$4500,  what  amount  does  each  company  pay  ? 

10.  A  man  insures  his  life  for  $10,000,  paying  $350 
a  year  in  advance,  and  dies  the  day  before  the  fifth  pre- 
mium is  due.  The  company  pays  his  widow  $10,000. 
How  much  has  the  company  lost  by  him,  if  the  interest 
gained  on  the  premiums  paid  amounts  to  $175  ? 

11.  A  merchant  shipped  a  cargo  to  London,  and  took  a 
policy  of  $100,800  at  3-J-%,  to  cover  both  the  cargo  and  the 
premium.     Find  the  value  of  the  cargo. 

Hint.  100%    of  policy  =  policy  (cargo  and  premium). 

2»\%  of  policy  =  premium. 
96£%  of  policy  =  cargo. 

12.  Three  companies  insure,  at  f  its  value,  a  building 
worth  $16,000.  The  first  company  takes  \  the  risk  at  f 
of  \of0  ;  the  second,  §  at  {  of  1%  ;  and  the  third,  the 
remainder  at  f  of  l°f0.     Find  the  total  premium. 

13.  S.  Williams  pays  $18.40  premium  for  insuring  his 
house  for  J  its  value  at  l-j-%.  What  is  the  value  of  his 
house  ? 

14.  Find  the  annual  premium  for  an  ordinary  life  policy 
of  $5000  issued  to  a  man  30  years  old,  if  the  rate  of  insur- 
ance is  1.93%. 

15.  What  is  the  annual  premium  for  an  ordinary  life 
policy  of  $12,000  issued  to  a  man  40  years  old,  if  the  rate 
of  insurance  is  2.661%  ? 


254  PERCENTAGE. 


Direct  Taxes. 

437.  Taxes  are  levied  for  the  support  of  the  govern- 
ment, for  the  support  of  schools,  and  for  other  purposes. 

438.  Direct  taxes  are  laid  on  the  person  or  on  the 
value  of  the  property  he  possesses. 

439.  A  poll  tax  is  a  tax  levied  upon  a  person,  and  a 
property  tax  is  a  tax  levied  on  property. 

440.  Assessors  are  officers  whose  duty  it  is  to  appraise 
the  property  belonging  to  each  person  to  be  taxed. 

441.  A  tax  collector  is  an  officer  who  collects  the  taxes. 
He  receives  a  salary,  or  a  per  cent  of  the  sum  collected. 

442.  The  treasurer  receives  the  money  collected,  and  is 
usually  paid  a  salary. 

443.  The  tax  to  be  raised  on  the  assessed  valuation  of 
property  is  the  total  amount  of  tax  voted,  diminished  by 
the  poll  taxes,  and  by  such  corporation  taxes  as  are  col- 
lected by  the  state  and  distributed  to  the  several  towns. 

A  tax  of  $18,000  is  levied  upon  a  town  which  contains 
800  polls,  assessed  at  $1.50  each,  and  which  lias  taxable 
property  assessed  at  $1,200,000.  The  town  receives  from 
the  state  $3600  as  its  share  of  corporation  taxes.  Find 
the  rate  of  taxation,  and  the  tax  paid  by  Mr.  Brown,  if  his 
property  is  assessed  at  $5960,  and  he  pays  for  one  poll. 

Solution.  The  amount  of  poll  taxes  =  800  X  §1.50  =  $1200;  and 
the  amount  from  state  and  polls  =  $3600  +  $1200  =  84800. 

The  amount  levied  on  property  =  $18,000  -  $4800     =  $13,200. 

The  rate  =  $13,200  -r  1,200,000  =  $0,011,  or  $11  on  $1000. 

To  provide  for  contingencies,  such  as  abatement  of  taxes,  cost  of 
collecting,  etc.,  the  assessors  make  the  rate  $12  on  $1000. 

Therefore,  Mr.  Brown's  property  tax  is  0.012  of  $5960  =  $71.52, 
and  his  total  tax  =  $71.52  +  $1.50  =  $73.02. 


PERCENTAGE. 


255 


444.  To  facilitate  the  making  of  taxes,  the  assessors 
usually  prepare  a  table  like  the  following,  which  is  com- 
puted at  12  mills  on  a  dollar  : 

Table. 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$0,012 

$10 

$0.12 

$100 

$1.20 

$1000 

$12.00 

2 

0.024 

20 

0.24 

200 

2.40 

2000 

24.00 

3 

0.036 

30 

0.36 

300 

3.60 

3000 

36.00 

4 

0.048 

40 

0.48 

400 

4.80 

4000 

48.00 

6 

0.060 

50 

0.60 

500 

6.00 

5000 

60.00 

6 

0.072 

60 

0.72 

600 

7.20 

6000 

72.00 

7 

0.084 

70 

0.84 

700 

8.40 

7000 

84.00 

8 

0.096 

80 

0.96 

800 

9.60 

8000 

96.00 

9 

0.108 

90 

1.08 

900 

10.80 

9000 

108.00 

445.  Example.  Find  by  the  above  table  the  tax  of 
Mr.  Gay,  who  pays  for  one  poll  at  $1.50,  and  has  property 
assessed  at  $6545. 


Solution. 


Tax  on 
Tax  on 
Tax  on 
Tax  on 


:  $72.00 

500  =      6.00 

40  s=      0.48 

5  =       0.06 


Tax  on  poll      =       1.50 
Total  tax      =  $80.04 

Exercise  116. 
Make  a  table  for  a  tax  rate  of  16  mills, 

1.  Find  the  tax  on  property  assessed 

2.  Find  the  tax  on  property  assessed 

3.  Find  the  tax  on  property  assessed 

4.  Find  the  tax  on  property  assessed 

5.  Find  the  tax  on  property  assessed 

6.  Find  the  tax  on  property  assessed 

7.  Find  the  tax  on  property  assessed 


and  : 

at  $7500. 
at  $4825. 
at  $9685. 
at  $10,727. 
at  $12,863. 
at  $16,458. 
at  $38,249. 


256  PERCENTAGE. 

8.  James  Brown  is  assessed  $2500  on  his  real  estate 
and  $5200  on  his  personal  property,  and  pays  for  two  polls 
at  $  1.50  each.  If  the  rate  is  $12.18  on  $1000,  what  is  his 
total  tax  ? 

9.  If  the  tax  rate  of  a  town  is  $12.25  on  $1000,  and 
the  amount  of  the  levy  $11,788.50,  what  is  the  assessed 
valuation  of  the  town  ? 

10.  If  the  assessed  valuation  of  a  town  is  $1,777,000, 
and  the  levy  is  $29,231.65,  what  is  the  rate  on  $1000  ? 

1 1.  What  sum  must  be  assessed  that  $15,000  may  remain 
after  paying  2%  commission  for  collecting  the  taxes  ? 

12.  For  building  a  schoolhouse  a  tax  of  $1857.60  was 
levied  upon  a  school  district,  assessed  valuation  $1,935,000. 
What  was  the  tax  on  property  assessed  at  $6250  ? 

13.  In  a  certain  town  there  are  1350  polls.  The  assessed 
valuation  of  the  real  estate  is  $713,250,  and  of  the  personal 
property  is  $738,954.  The  poll  tax  is  $2  per  poll,  and 
the  tax  on  property  is  1£%.  Only  96%  of  the  property 
tax  can  be  collected,  and  the  collector  is  paid  2£%  of  the 
amount  collected.  How  much  does  the  town  receive  from 
the  taxes  ?  How  much  does  the  collector  receive  for  his 
services  ? 

Indirect  Taxes. 

446.  Indirect  Taxes  are  taxes  levied  upon  articles  of 
merchandise. 

447.  Indirect  taxes  are  of  two  kinds  :  Customs  or 
Duties,  taxes  levied  on  certain  imported  goods  ;  Excises 
or  Internal  Revenue,  taxes  levied  on  certain  domestic 
goods. 

Note.  These  are  called  Indirect  Taxes,  because,  while  paid  to  the 
government  by  the  person  in  whose  possession  the  goods  are  first 
found,  this  tax  forms  a  part  of  the  price  finally  paid  by  the  consumer, 
and  is,  therefore,  a  tax  paid  by  him. 


PERCENTAGE.  257 

448.  Duties  are  of  two  classes  :  Specific  and  Ad  Valorem. 

449.  Specific  Duties.  A  specific  duty  is  a  definite  sum 
levied  on  each  unit  by  which  the  article  is  measured  or 
weighed,  without  regard  to  the  value  of  the  unit. 

450.  Ad  Valorem  Duties.  An  ad  valorem  duty  is 
levied  as  a  certain  per  cent  of  the  cost  of  the  article  in 
the  country  where  purchased.  The  cost  of  the  article 
includes  the  cost  of  transportation,  commissions,  etc. 

451.  On  merchandise  imported  at  a  specific  duty  the 
following  allowances  are  made  : 

Tare  is  an  allowance  for  the  weight  of  the  box,  cask, 
bag  or  other  wrapping. 

Breakage  is  an  allowance  of  a  certain  per  cent  on  liquors 
in  bottles ;  also  on  glassware  and  china. 

Leakage  is  an  allowance  of  a  certain  per  cent  on  liquors 
in  barrels  or  casks. 

Duties  are  computed  on  the  sum  left  after  all  deductions 
are  made. 

Note.     The  long  ton  only  is  used  in  computing  duties. 

452.  Excises  are  taxes  or  licenses  for  the  manufacture 
or  sale  of  certain  domestic  articles,  as  tobacco,  whiskey, 
beer,  etc. 

Exercise  117. 

In  the  following  examples  the  rates  of  duty  are  taken  from  the 
Dingley  Tariff  Law,  in  effect  July  24,  1897. 

1.  What  is  the  duty  at  2  J-  cents  a  pound  on  320  boxes 
of  raisins  each  containing  40  pounds  ? 

2.  What  is  the  duty  at  6  cents  a  gallon  on  420  hhd.  of 
best  molasses  of  63  gal.  each  ? 

3.  What  is  the  duty  at  $4  a  dozen  bottles  on  50  cases 
of  champagne,  each  containing  24  pint  bottles,  if  breakage 
of  5  of0  is  allowed  ? 


258  PERCENTAGE. 

4.  Find  the  duty  on  150  gross  of  spectacles,  cost  price 
$1.20  a  dozen ;  specific  duty  45  cents  a  dozen,  breakage 
allowed  2£%  ;   an<l  20%  ad  valorem. 

5.  Find  the  duty  on  100  shotguns,  cost  price  $8.50 
each  ;   specific  duty  of  $4  each,  and  15%  ad  valorem. 

6.  Find  the  duty  at  $1  per  M  on  12,500  ft.  of  white- 
wood  boards,  planed  on  one  side,  if  an  additional  duty  of 
50  cents  per  M  is  collected  for  each  side  planed. 

7.  Find  the  duty  on  500  boxes  of  cigars,  gross  weight 
475  lb.,  tare  40%,  costing  82-j-  cents  per  box  in  Havana. 
Specific  duty  $4.50  per  pound ;    and  25%  ad  valorem. 

8.  Find  the  duty  on  400  pairs  of  woolen  blankets,  cost 
price  $1.75  per  pair;  weighing  7£  lb.  per  pair,  tare  5%. 
Specific  duty  33  cents  per  pound,  ad  valorem  40%. 

9.  Find  the  duty  on  12  boxes  of  skein  silk,  each  box 
weighing  40  lb.;  cost  price  $2,125  per  pound,  tare  10%. 
Specific  duty  50  cents  per  pound,  ad  valorem  15%. 

10.  Find  the  duty  on  150  gross  of  clay  tobacco  pipes, 
cost  price  55  cents  a  gross.  Specific  duty  15  cents  a  gross, 
and  25%  ad  valorem. 

11.  A  New  York  merchant  bought  in  London  400  gal. 
of  cologne  at  $1.25  a  gallon,  and  commission  and  other 
expenses  amounted  to  $56.25.  At  what  price  per  pint 
must  he  sell  the  cologne  to  gain  40%  on  the  cost,  if  he 
paid  a  specific  duty  of  60  cents  a  gallon,  and  an  ad  valorem 
duty  of  45%  ? 

12.  Find  the  duty  on  750  lb.  of  glue,  cost  price  40  cents ; 
specific  duty  of  15  cents  a  pound,  tare  2% ;  and  ad  valorem 
duty  of  25%. 

13.  A  Boston  merchant  bought  in  Sheffield  50  gross 
of  razors  at  a  net  price  of  $4.25  a  dozen.  At  what  price 
per  dozen  must  he  sell  the  razors  to  gain  33£%  on  the  net 
cost,  if  he  paid  a  specific  duty  of  $1.75  a  dozen,  and  an  ad 
valorem  duty  of  20%  ? 


CHAPTER   XIII. 

INTEREST  AND  DISCOUNT. 

Simple  Interest. 

453.  Interest  is  money  paid  for  the  use  of  money. 

454.  Principal.     The  sum  loaned  is  the  principal. 

455.  Rate  of  Interest.  The  rate  of  interest  is  a  specified 
per  cent,  called  rate  per  cent,  of  the  principal  for  one  year. 

456.  Amount.  The  sum  of  the  principal  and  interest  is 
the  amount. 

Note.  In  the  applications  of  percentage  considered  in  Chapter 
XII  no  reference  has  been  made  to  time.  In  all  applications  of  per- 
centage that  relate  to  the  use  of  money  the  elements  of  time  and  rate 
are  considered. 

457.  Example.  Find  the  interest  on  $1024  at  5£%  for 
2  yr.  8  mo. 

Solution.     2  yr.  8  mo.  =  2£  yr. 

Interest  on  $1024  for  1  yr.    =a  &*%  of  $1024  =  $56.32. 

Interest  on  $1024  for  2f  yr.  =  2f  X  $56.32    =  $150.19. 

Exercise  118. 
Find  the  interest  on  : 

1.  $125.65  for  1  mo.  at  6%.     3.  $1296.50  for  2  mo.  at  5£%. 

2.  $1165  for  3  yr.  at  5%.        4.  $630.50  for  3  yr.  at  4%. 

5.  $231.50  for  3  yr.  8  mo.  at  4£%. 

6.  $580.40  for  2  yr.  4  mo.  at  6%. 

7.  $285.85  for  1  yr.  7  mo.  at  4%. 

8.  $1275.35  for  3  yr.  2  mo.  at  3|%. 


260 


INTEREST    AND    DISCOUNT. 


Six  Per  Cent  Method. 

458.  The  interest  at  six  per  cent  on  one  dollar  for  one 
year  is  6  cents  ;  for  one  month  is  T^  of  6  cents,  or  £  cent ; 
for  one  day  is  ^  of  \  cent,  or  ^  cent,  that  is,  £  mill. 

The  interest  at  six  per  cent  on  any  number  of  dollars  is 
that  number  times  the  interest  on  one  dollar.     Hence, 

459.  To  Find  the  Interest  at  Six  Per  Cent  on  any  Prin- 
cipal for  Time  expressed  in  Years,  Months,  and  Days, 

Multiply  six  cents  by  the  number  of  years,  one  half  cent  by 
the  number  of  months,  one  sixth  mill  by  the  number  of  days. 
Take  the  sum  of  these  products,  and  multiply  this  sum  by 
the  number  of  dollars  in  the  principal. 


460.   Example. 
9  mo.  20  dy.,  at  6 

Int.  on  ft  1. 

For  3  yr.    =    3  X  §0.06 
For  9  mo.   =    9  X    0.005 
For  20  dy.  =  20  X    0.000; 
For  3  yr.  9  mo.  20  dy. 


Find  the  interest  on  $320  for  3  yr. 


=  $0.18 

Int.  on  $120. 
$0,228* 
320 
106* 
456 
684 
$73.07 

119. 

Condensed  Work. 

3  yr.  9  mo.  20  dy. 

s=    0.045 
=    0.003* 

$0.18   0.045  0.003* 
0.045 

=  $0,228* 

0.003* 

$0,228* 
320 

Exercise 

Find  the  interest  at  6%  on  : 

1.  $744.20  for  3  yr.  6  mo.  18  dy. 

2.  $625.44  for  6  yr.  7  mo.  12  dy. 

3.  $124.87  for  2  yr.  10  mo.  16  dy. 

4.  $847.64  from  Jan.  12,  1896  to  Aug.  7,  1899. 

5.  $84.84  from  Mar.  22,  1895  to  Jan.  1,  1898. 

6.  $1248.27  from  Apr.  7,  1894  to  May  17,  1897. 

461.    Short  time  notes  generally  run  for  1,  2,  3,  or  4 
months  ;   or  for  30,  60,  or  90  days. 


INTEREST   AND   DISCOUNT.  261 

462.  To  Find  Interest  at  6%  for  Time  in  Months, 

Move  the  decimal  point  in  the  principal  two  places  to  the 
left,  and  multiply  by  half  the  number  of  months. 

463.  To  Find  Interest  at  6%  for  Time  in  Days, 

Move  the  decimal  point  in  the  principal  three  places  to  the 
left,  and  multiply  by  one  sixth  the  number  of  days. 

Thus,  the  interest  at  6%  on  $600  for  4  mo.  is  2  X  $6.00,  or  $12 ; 
and  for  30  dy.  is  5  X  $0,600,  or  $3. 

Exercise  120. 
Find  the  interest  at  6  %  on  : 

1.  $1278.75  for  1  mo.;    2  mo.;   3  mo.;   4  mo. 

2.  1 2265.50  for  1  mo. ;    2  mo.;    3  mo.;    4  mo. 

3.  $1840.25  for  30  dy.;    60  dy.;   90  dy. 

4.  $1946.75  for  30  dy.;    60  dy.;    90  dy. 

Six  Per  Cent  Method  for  Other  Rates. 

464.  Example.  Find  the  interest  on  $73.42  for  5  yr. 
8  mo.  16  dy.,  at  l%°f0. 

Solution.     5  yr.    8  mo.    16  dy.  $0,342$- 

$0.30     0.04     0.002*  73.42 

0.04  6)  $25. 158 

0.002*  $4,193 

$0,342$  Multiply  by  7£ 

$31.45 

465.  To  Find  Interest  at  Rates  other  than  6%, 

Divide  the  interest  at  six  per  cent  by  six  and  multiply  the 
quotient  by  the  given  rate.     Or, 

Take  such  a  part  of  the  interest  at  six  per  cent  as  the 
given  rate  is  of  six  per  cent. 

For  7%,  we  add  \  of  the  interest  at  6%  to  the  interest  at  6% ;  for 
7|%,  we  add  \ ;  for  8%,  we  add  | ;  for  9%,  we  add  \ ;  for  5%,  we  sub- 
tract \ ;   for  4|%,  we  subtract  \ ;   for  4%,  we  subtract  \. 


262  INTEREST   AND   DISCOUNT. 

Exercise  121. 
Find  the  interest  on  : 

1.  $680.40  for  2  yr.  4  mo.  6  dy.,  at  6%. 

2.  $25.62  for  30  dy.,  at  6%. 

3.  $85.85  for  1  yr.  7  mo.  21  dy.,  at  6%. 

4.  $1100  for  3  yr.  4  mo.,  at  5%. 

5.  $1275  for  3  yr.  2  mo.  15  dy.,  at  8%. 

6.  $475.16  for  27  dy.,  at  4£%. 

7.  $1290.50  for  60  dy.,  at  6%. 

8.  $125  for  1  yr.  2  mo.  2  dy.,  at  9%. 

9.  $250.80  for  10  mo.  10  dy.,  at  3|%. 

10.  $258.85  from  Mar.  6  to  June  24,  at  5%. 

11.  $380  for  2  yr.  11  mo.  27  dy.,  at  4£%. 

12.  $475.05  for  1  yr.  9  mo.  14  dy.,  at  7T\%. 

13.  $725.40  for  11  mo.  24  dy.,  at  5±%. 

14.  $680.50  for  2  yr.  6  dy.,  at  5%. 

15.  $630.50  for  90  dy.,  at  6%. 

16.  $547.60  from  Feb.  20  to  Dec.  5,  at  6£%. 

17.  $875  from  May  5, 1897  to  June  21, 1898,  at  5£%. 

18.  $758.50  from  Jan.  5  to  July  1,  at  4£%. 

19.  $342.42  from  Feb.  5,  1897  to  Mar.  15,  1899,  at  7%. 

20.  $540  from  Mar.  5  to  Sept.  21,  at  3£%. 
Find  the  amount  of  : 

21.  $431.50  for  2  yr.  8  mo.,  at  4J%. 

22.  $476.50  from  July  5,  1897  to  Feb.  9,  1898,  at  4%. 

23.  $319.20  from  Apr.  7  to  Aug.  31,  at  3£%. 

24.  $6460  from  June  15,  1897  to  May  7,  1899,  at  4J<&. 

25.  $150  from  Aug.  5,  1897  to  Mar.  17,  1899,  at  7%. 

26.  $527.20  from  Jan.  1  to  Nov.  20,  at  4£%. 

27.  $1250  from  Nov.  15,  1897  to  Mar.  1,  1898,  at  5%. 

28.  $624.36  from  Mar.  5  to  Dec.  20,  at  7^%. 

29.  $12,260  from  May  6  to  Oct.  24,  at  3|%. 

30.  $11,216  from  Oct.  20  to  Dec.  31,  at  1%  a  month. 


INTEREST   AND   DISCOUNT.  263 


Other  Interest  Problems. 

466.  In  interest  problems  four  elements  are  considered  : 
principal,  rate  per  cent,  time,  and  interest  or  amount. 

In  the  problems  already  considered  the  first  three  ele- 
ments are  given  to  find  the  fourth. 

If  p  stands  for  the  principal,  r  for  the  rate  per  cent,  t  for 
the  time  expressed  in  years,  i  for  the  interest,  and  a  for  the 
amount,  we  have 

1.   prt  =  i.         2.  p  -\-prt  =  a. 

In  either  equation,  if  three  of  the  elements  are  given,  the 
other  can  be  found. 

467.  Examples.  1.  What  principal  will  in  2  yr.  7  mo. 
24  dy.  produce  $1006.47  interest,  at  4|%  ? 

Solution.     Divide  by  rt  both  sides  of  the  equation 

prt  =  i; 

then  p  =  —  • 

rt 

HereZ,  2  yr.  7  mo.  24dy.,  =  2.65yr.;  r,  4£%,  =  0.045;  i  =  $1006.47. 

Hence'  ^  =  $0.0140506x4J.65  =  $8440- 

2.'  What  principal  will  in  3  yr.  6  mo.  amount  to  $748.41, 
at  4%  ? 

Solution.     Divide  by  1  +  rt  both  sides  of  the  equation 
p  +  prt  =  a ; 
a 


then  p 


1  +rt 


Here  a  =  $748.41 ;   r,  4%,  =  0.04  ;   *,  3  yr.  6  mo.,  =  3.5  yr.;   and 
1  '+  rt  =  1  +  0.04  X  3.5  =  1  +  0.14  =  1.14. 

Hence,  p  -  $  ^yrr^  =  $656. 50. 


264  INTEREST    AND    DISCOUNT. 

3.  At  what  rate  per  cent  will  $8440  produce  $1006.47 
interest  in  2  yr.  7  mo.  24  dy.? 

Solution.     Divide  by  pt  both  sides  of  the  equation 
prt  =  i; 

i 
then  r  =  —  • 

pt 

Here  i  =  $1006.47  ;  p  =  $8440  ;   t,  2  yr.  7  mo.  24  dy.,  =  2.65  yr. 

1006.47   .       nnA. 

HenCG'  T  =  8440  X  2.65  -0045' 

Therefore,  the  rate  required  is  4£%. 

4.  In  what  time  will  the  interest  on  $8440  at  4£% 
amount  to  $1006.47  ? 

Solution.     Divide  by  pr  both  sides  of  the  equation 

prt  =  i; 

i 
then  t  =  — 

pr 

Here  i  =  $1006.47  ;  p  =  $8440  ;  r,  4*%,  =  0.045. 

.  1006.47  onc 

HeDCe'  '  =  8440  X  0.045  =  2M' 

Therefore,  the  time  required  is  2.65  yr.,  or  2  yr.  7  mo.  24  dy. 

468.  From  these  examples  we  have  the  following  rule 
for  finding  the  principal,  rate,  or  time : 

Divide  the  given  interest  (or  amount)  by  the  interest  (or 
amount)  obtained  when  the  required  element  is  represented  by 
a  unit. 

The  unit  of  this  rule  is  $1  in  finding  the  principal ;  1%  in  finding 
the  rate  ;   1  yr.  in  finding  the  time. 

Exercise  122. 
Find  the  rate  per  cent : 

1.  When  the  interest  on  $326  for  15  yr.  is  $220.05. 

2.  When  the  interest  on  $745  for  18  yr.  is  $603.45. 

3.  When  $980  amounts  to  $1016.75  in  9  mo. 


INTEREST    AND    DISCOUNT.  265 

«  Find  the  rate  per  cent : 

4.  When  the  interest  on  $470.50  is  $141.15  for  5  yr. 

5.  When  $3631.25  amounts  to  $3715.98  for  7  mo. 

6.  When  the  interest  on  $997.75  is  $199.55  for  5  yr. 
4  mo. 

7.  When  $350  amounts  $406.70  for  3  yr.  7  mo.  6  dy. 

8.  When  the  interest  on  $6875  is  $68.75  for  90  dy. 

9.  When  the  interest  on  $642  is  $10.70  for  5  mo. 

10.  When  the  interest  on  $8432  for  2  yr.  7  mo.  23  dy. 
is  $1339.28. 

11.  When  a  sum  of  money  is  doubled  in  14  yr. 

12.  When  an  investment  for  4  yr.  2  mo.  produces  a  sum 
equal  to  /¥  of  the  capital. 

13.  When  an  investment  for  3  yr.  1  mo.  15  dy.  produces 
a  sum  equal  to  ^  of  the  capital. 

Find  the  time  in  which  the  : 

14.  Interest  on  $450  will  amount  to  $72,  at  4%. 

15.  Interest  on  $487.50  will  amount  to  $39,  at  4%. 

16.  Interest  on  $238.75  will  amount  to  $64.46,  at  4£%. 

17.  Sum  of  $1587.75  will  amount  to  $1611.68,  at  5£%. 

18.  Sum  of  $1  will  double  itself  at  4%. 

19.  Sum  of  $10  will  amount  to  $17,  at  6%. 

20.  Sum  of  $502.67  will  amount  to  $578.07,  at  4£%. 

21.  Interest  on  $537.50  will  amount  to  $80.62,  at  4%. 

22.  Interest  on  $6875  will  amount  to  $75.05,  at  4£%. 

23.  Interest  on  $8520  will  amount  to  $1746.60,  at  6%. 
Find  the  principal  that  will  : 

24.  Produce  $90  interest  in  3  yr.,  at  4%. 

25.  Produce  $63  interest  in  3  yr.,  at  6£%. 

26.  Produce  $100  interest  in  8  yr.  6  mo.,  at  5%. 

27.  Produce  $1746.60  interest  in  3  yr.  5  mo.,  at  6%. 

28.  Produce  $12  interest  in  7  mo.,  at  5%. 

29.  Produce  $50  interest  in  228  dy.,  at  4£%. 


266  INTEREST   AND   DISCOUNT. 

Find  the  principal  that  will : 

30.  Produce  $  1339.28  interest  in  2  yr.  7  mo.  24  dy.,  at 
6%. 

31.  Produce  $1312.65  interest  in  2  yr.  3  mo.,  at  6%. 

32.  Produce  $750  interest  in  3  yr.  8  mo.,  at  5%. 

33.  Amount  to  $840  in  3  yr.,  at  4%. 

34.  Amount  to  $90,113.84  in  2  yr.  6  mo.,  at  4£%. 

35.  Amount  to  $6000  in  21  dy.,  at  5%. 

36.  Amount  to  $297.60  in  8  mo.,  at  6%. 

37.  Amount  to  $6378.75  in  1  .yr.  1  mo.,  at  5%. 

38.  Amount  to  $21,047.95  in  1  yr.  7  mo.  21  dy.,  at  4£%. 

39.  Amount  to  $185.09  in  2  yr.  3  mo.  18  dy.,  at  5%. 

40.  Amount  to  $659.40  in  2  yr.  11  mo.  15  dy.,  at  6%. 

41.  Amount  to  $94,375.16  in  2  yr.  7  mo.  24  dy.,  at  4£%. 

42.  Amount  to  $10,266.60  in  3  yr.  5  mo.,  at  6%. 

43.  Find  the  interest  on  $195  for  2  yr.  2  dy.,  at  6£%. 

44.  At  what  rate  per  cent  will  $1025.20  produce  $25.72 
interest  in  4  mo.  9  dy.? 

45.  The  principal  is  $653;   the  interest,   $5.52;   the 
rate,  8%.     Find  the  time. 

46.  Find  the  amount  of  $520  for  2  mo.  3  dy.,  at  4£%. 

47.  What  sum  bearing  interest  at  \\°j0  will  yield  an 
annual  income  of  $1000  ? 

48.  In  what  time  will  $4000  amount  to  $4625,  at  5£%  ? 

49.  At  what  rate  per  cent  will   $3000  produce  $250 
interest  in  1  yr.  10  mo.  7  dy.? 

50.  Find  the   interest   on   $1721.84    from    April  1  to 
Nov.  12,  at  4£%. 

51.  How  long  must  $3904.92  be  on  interest  to  amount 
to  $4568.76,  at  5%  ? 

52.  Find  the  interest  on  $137.60  from  July  3  to  Dec.  12, 
at  7A%. 

53.  Find  the  interest  on  $680.20,  at  7£%,  for  73  dy., 
reckoning  365  dy.  for  a  year. 


INTEREST   AND   DISCOUNT.  267 

Promissory  Notes. 

469.  Promissory  Notes.  A  written  promise  to  pay  a 
specified  sum  of  money  on  demand  or  at  a  specified  time  is 
called  a  promissory  note,  or  simply  a  note. 

A  note  may  be  made  payable  to  bearer,  to  a  person  named  in  the 
note,  or  to  the  person  named  or  his  order. 

The  maker  or  drawer  is  the  person  who  signs  the  note. 

The  payee  or  drawee  is  the  person  to  whom  the  note  is  payable. 

The  holder  of  a  note  is  the  person  who  has  lawful  possession  of  it. 

The  face  of  a  note  is  the  sum  of  money  named  in  it. 

A  negotiable  note  is  a  note  that  can  be  sold  and  transferred  to  any 
one  else  by  the  holder.  A  note  is  not  negotiable  unless  made  payable 
to  bearer,  or  to  the  order  of  the  person  named  in  the  note. 

470.  Forms  of  Notes. 

1.  $875.25.  Boston,  Mass.,  Nov.  1,  1897. 
Four  months  after  date,  I  promise  to  pay  Benjamin  Parker 

Three  Hundred  Seventy-five  and  T2^  Dollars,  with  interest  at 
5%,  for  value  received.  James  Overton. 

2.  $375.25.  Boston,  Mass.,  Nov.  1,  1897. 
Four  months  after  date,  I  promise  to  pay  Benjamin  Parker, 

or  bearer,  Three  Hundred  Seventy-five  and  t2q5^  Dollars,  with 
interest  at  5°J0,for  value  received.  James  Overton. 

3.  $375.25.  Boston,  Mass.,  Nov.  1,  1897. 
Four  months  after  date,  I  promise  to  pay  Benjamin  Parker, 

or  order,  Three  Hundred  Seventy-five  and  y2^  Dollars,  with 
interest  at  5°]0,for  value  received.  James  Overton. 

4.  $375.25.  Boston,  Mass.,  Nov.  1,  1897. 
On  demand,  I  promise  to  pay  Benjamin  Parker,  or  order, 

Three  Hundred  Seventy-five  and  -fife  Dollars,  with  interest 
at  5%,  for  value  received.  James  Overton. 


268  INTEREST   AND   DISCOUNT. 

471.  An  endorser  of  a  note  is  a  person  who  writes  his 
name  on  the  back  of  a  note. 

An  endorser  of  a  note  becomes  responsible  for  its  payment,  unless 
he  writes  above  his  signature  the  words  without  recourse. 

A  note  that  contains  the  words  or  bearer  can  be  collected  by  any 
one  who  has  lawful  possession  of  it. 

A  note  that  contains  the  words  or  order  becomes  payable  to  bearer 
when  the  payee  merely  writes  his  name  across  the  back.  This  is 
called  an  Endorsement  in  blank. 

In  form  3,  if  Benjamin  Parker  sells  the  note  to  James  Whitney  he 
writes  on  the  back  Pay  to  the  order  of  James  Whitney  and  signs  his 
name,  or  else  he  simply  endorses  in  blank. 

Notes  that  contain  the  words  on  demand  are  called  Demand  Notes, 
and  are  due  whenever  payment  is  demanded;  see  form  4.  Other 
notes  are  called  Time  Notes.  The  day  on  which  a  time  note  is  legally 
due  is  called  the  day  of  its  maturity. 

Notes  containing  the  words  with  interest  bear  interest  from  date 
until  paid.  Notes  not  containing  these  words  begin  to  bear  interest 
from  the  time  they  are  due  at  the  legal  rate,  if  not  paid  when  due. 

Legal  rate  of  interest  is  the  rate  established  by  law  in  the  state 
where  the  note  is  made. 

Notes  should  contain  the  words  value  received,  otherwise  it  may  be 
necessary  to  prove  the  note  was  given  for  a  valuable  consideration. 
The  face  of  a  note,  except  the  cents,  should  be  expressed  in  words. 


Exercise  123. 

Find  the  day  of  maturity,  and  amount  due,  having  given  : 

Face  of  Note. 

Date  of  Note. 

Time. 

Rate  of  Int. 

1. 

$530.25, 

Jan.  12,  1897, 

60  dy., 

6%. 

2. 

$687.45, 

Mar.  22,  1897, 

90  dy., 

5%. 

3. 

$286.75, 

Aug.  5,  1897, 

4  mo., 

4%. 

4. 

$944.40, 

Oct.  20,  1897, 

3  mo., 

H%- 

5. 

$1262.72, 

Oct.  5,  1897, 

30  dy., 

mo. 

6. 

$1875.44, 

Dec.  16,  1897, 

6  mo., 

4%. 

7. 

$1521.87, 

Apr.  30,  1897, 

1  mo., 

6%- 

8. 

$2849.65, 

May  22,  1897, 

2yr., 

3£%. 

9. 

$1968.10, 

July  10,  1897, 

2  mo., 

4i%- 

INTEREST   AND    DISCOUNT.  269 

Find  the  amount  due  Dec.  3,  1898,  on  the  following 
demand  notes  : 

10.  $875.18.  Concord,  K  H.,  May  10,  1897. 

On  demand,  I  promise  to  pay  George  H.  Chick,  or  order, 
Eight  Hundred  Seventy-five  and  -J^  Dollars,  with  interest 
at  5%.     Value  received.  Frederick  D.  Sibley. 

11.  $642.75.  Lakewood,  N.  J.,  Oct.  25,  1897. 

On  demand,  I  promise  to  pay  Harry  Jones,  or  order,  Six 
Hundred  Forty-two  and  Ty^  Dollars,  with  interest  at  4-J-%. 
Value  received.       -  George  B.  Atkins. 

12.  $1286.50.  Atlanta,  Ga.,  Apr.  22,  1897. 

On  demand,  I  promise  to  pay  Clarence  E.  Garland,  or 
order,  Twelve  Hundred  Eighty-six  and  T5^  Dollars,  with 
interest  at  h\°f0.     Value  received.  Kobert  Page. 

13.  $2548.25.  St.  Paul,  Minn.,  June  17,  1897. 
On  demand,   I  promise  to  pay  Fred  Lacey,  or  order, 

Twenty-five  Hundred  Forty-eight  and  j%5^  Dollars,  with 
interest  at  7%.    Value  received.     William  P.  Wissman. 

14.'  $418.33.  Oakland,  Cal.,  Dec.  23,  1897. 

On  demand,  I  promise  to  pay  Albert  J.  Farnham,  or 
order,  Four  Hundred  Eighteen  and  T3^  Dollars,  with 
interest  at  4^%.     Value  received.       Austin  C.  Wiggin. 

15.    $7486.45.  Watertown,  Ia.,  Apr.  16,  1898. 

On  demand,  I  promise  to  pay  Harry  D.  Smith,  or  order, 
Seven  Thousand,  Four  Hundred  Eighty-six  and  fify  Dollars, 
with  interest  at  5%,  Frank  j.  Leavitt, 


270  INTEREST   AND   DISCOUNT. 

Bank  Discount. 

472.  Bank  discount  is  a  deduction  from  the  amount  of 
a  note  or  draft  for  cashing  it  before  it  is  due. 

473.  The  term  of  discount  is  the  time  from  the  day  of 
discounting  the  note  to  and  including  the  day  of  maturity 
of  the  note. 

474.  Days  of  Grace.  Three  days  are  allowed  in  addi- 
tion to  the  time  stated  in  the  note  "before  the  note  is  legally 
due,  unless  the  note  contains  the  words  without  grace,  or  is 
made  in  states  that  have  abolished  days  of  grace  by  statute. 

475.  A  protest  is  a  notice  in  writing  by  a  notary  public 
to  the  endorsers  that  a  note  has  not  been  paid.  If  a  note 
is  not  protested  at  the  end  of  the  day  of  its  maturity  the 
endorsers  are  released  from  their  obligation. 

476.  Bank  discount  is  the  interest  for  the  term  of  dis- 
count at.  a  specified  rate  on  the  amount  of  the  note. 

477.  If  a  note  draws  interest,  the  amount  of  the  note  is 
the  face  and  interest. 

478.  Exchange.  A  charge  is  sometimes  made  for  col- 
lecting, if  the  place  of  payment  of  the  note  or  draft  is  not 
the  place  of  discount.     This  charge  is  called  exchange. 

The  rate  of  exchange  is  a  number  of  cents  on  $1000  for 
large  amounts ;  and  for  small  amounts  from  \  to  £  of  1% 
of  the  face  value  of  the  note  or  draft,  a  fraction  of  $100 
being  reckoned  as  $100. 

Note.  The  rate  of  exchange  is  never  higher  than  the  cost  of  send- 
ing the  money  by  express. 

479.  A  bank  draft  is  a  written  order  from  a  bank 
requesting  another  bank  to  pay  a  specified  sum  of  money 
to  the  order  of  the  person  named  in  the  draft.     Thus, 


INTEREST   AND   DISCOUNT.  271 


No.  27861 

<§'\x*t  (Uationae  (ganH  of  (guffafo* 

Buffalo,  N.Y.     &e&.  /5,  1897. 
Pay  to  the  order  of     fo-hn,  /?.  RUkcvuU      $/ 72.1*5. 
Q.n&  Jvzi'vigUl&cI  Qs&v-&nZu-tw~o  oavcL ^  Dollars. 

TO  THE  PARK  NATIONAL  BANK,  \  g,  &.  l^t&iwmA, 

NEW  YORK  CITY.  !  Cashier, 


480.   A  check  is  a  draft  upon  a  bank  where  the  maker 
(or  drawer)  has  money  deposited.     Thus, 


No.  89 

Boston, 

Mass., 

Jbee,.  /5,  1897. 

^First  National  &ank  of  Sosfort. 

Pay  to  the  order  of  fahyv  Jba& — 

$8  6. £5. 

(olg/itu- 

&u?o  cwul> 

jf0  Dollars. 

a. 

cf.    ^WAJbhe*. 

481.   Certificate  of  Deposit.    A  receipt  for  money  depos- 
ited jn  a  bank  is  called  a  certificate  of  deposit.     Thus, 


No.  28642  Certificate  of  Deposit. 

Boston,  Mass.,  h&&.  /5,  1897, 
yaAru&fa  @s.  jC&CLwJX/  has  deposited r  in  this  Bank 

3\/u&  hwyicOb&cl Dollars 

payable  to  rw&  order   in  current  funds  on  the  return  of  this  cer- 
tificate properly  endorsed. 

$600 .  RL&kcVuL  Ro&,    Cashier. 


272  INTEREST    AND    DISCOUNT. 

482.  A  certified  check  is  a  check  upon  the  face  of  which 
the  cashier  of  the  bank  has  stamped  the  word  Certified, 
with  the  date,  the  name  of  the  bank,  and  has  written  his 
signature  as  cashier.  The  bank  is  then  responsible  for  its 
payment. 

483.  A  commercial  draft  is  in  effect  a  letter  from  one 
person  to  another  requesting  him  to  pay  a  stated  sum  of 
money  to  the  bank  named  in  the  draft.  The  name  of  the 
person  on  whom  the  draft  is  drawn  is  written  at  the  lower 
left-hand  corner  of  the  draft.  These  drafts  are  sent 
through  banks  instead  of  through  the  mail ;  and  are  pay- 
able at  sight,  or  at  a  specified  time  after  sight. 

Commercial  drafts  are  employed  by  creditors  to  demand 
payments  and  collect  debts  through  banks. 


No.  93  Springfield,  Mass.,  Jbz&.  /5,  i%97. 

At  sight  pay  to  the  order  of 

€be  gjcconti  National  3Sanfe  of  ^prinfffielu       f/OO.™ 

€n&  kwncU&cL ■ Dollars. 

To     1&.  /if.  &owiojynve,,  *\      <?        /j-    is 

&ZO-.  Jr.  /TCvmML. 


/q6  £ve,wi  tit.,  cfi&w- 


. 


484.  When  the  person  on  whom  a  time  draft  is  drawn 
accepts  the  draft,  he  writes  in  red  ink  the  word  Accepted 
with  the  date  across  the  face  of  the  draft  and  signs  his 
name.  The  draft  is  then  called  an  acceptance,  and  the 
acceptor  is  responsible  for  its  payment. 

Note.  An  acceptance  is  like  a  promissory  note,  the  acceptor  and 
the  maker  of  the  draft  taking  the  place  respectively  of  the  maker  and 
the  endorser  of  the  note. 


INTEREST   AND   DISCOUNT.  273 

485.  The  proceeds  or  avails  of  a  note  or  draft  is  the 
amount  of  the  note  or  draft  less  the  discount  and  exchange. 

486.  Examples.  Find  the  day  of  maturity,  the  time  to 
run,  the  discount,  and  the  proceeds  of  the  following  notes : 

1.  $680.  Manchester,  N.  H.,  Oct.  5,  1897. 
Sixty  days  after  date,  without  grace,  I  promise  to  pay 

F.  R.  Thompson,  or  order,  Six  Hundred  Eighty  Dollars, 
value  received. 

Payable  at  the  Amoskeag  National  Bank. 

Discounted  at  6%,  Oct.  14.  B.  F.  Shaw. 

Solution.  Counting  60  days  from  Oct.  5,  we  have  26  days  in  Oct. , 
30  days  in  Nov.,  and  4  days  in  Dec.     Therefore,  the  note  is  due  Dec.  4. 

The  time  to  run  (from  Oct.  14,  the  day  of  discount)  is  17  days  in 
Oct.,  30  days  in  Nov.,  and  4  days  in  Dec,  or  51  days. 

The  discount  is  the  interest  on  $680  for  51  days  at  6% ;  or  8£  X 
$0,680  =  $5.78  (§  463). 

The  proceeds  is  $680  —  $5.78  =  $674.22. 

2.  $840.  Manchester,  N.  H.,  Oct.  5,  1897. 
Three  months  from  date,  I  promise  to  pay  E.  A.  Jackson, 

or  order,  Eight  Hundred  Forty  Dollars,  value  received,  with 
interest  at  six  per  cent. 

Payable  at  the  Amoskeag  National  Bank. 

Discounted  at  6%,  Nov.  2, 1897.      G-.  A.  Batchelder. 

SSlution.  The  interest  on  the  note  for  3  mo.  3  dy.  is  840  X  $0.0156 
=  $13.02  ;  and  the  amount  of  the  note  is  $840  +  $13.02  =  $853.02. 

The  day  of  maturity  is  Jan.  8,  1898. 

The  time  to  run  is  28  days  in  Nov.,  31  days  in  Dec,  and  8  days  in 
Jan.,  or  67  days. 

The  discount  on  $853.02  for  67  days  at  6%  is  11|  X  $0.85302,  or 
$9.53. 

The  proceeds  is  $853.02  —  $9.53  =  $843.49. 

Note.  In  Boston,  and  in  many  other  places,  when  the  time  a 
note  has  to  run  is  expressed  in  months,  the  term  of  discount  is  com- 
puted for  this  number  of  months,  and  not  for  the  exact  number  of 
days  contained  in  the  months. 


274  INTEREST   AND   DISCOUNT. 

Exercise  124. 

Find  the  day  of  maturity,  the  time  to  run,  the  discount, 
and  the  proceeds  of  the  following  notes,  without  grace  : 

1.  $750.  New  York,  Jan.  1,  1897. 
Four  months  from  date,  I  promise  to  pay  to  the  order  of 

James  Fay  Seven  Hundred  Fifty  Dollars,  value  received. 
Payable  at  the  National  Bank  of  the  Republic. 
Discounted  at  5%,  Jan.  12.  John  Pray. 

2.  $4325.50.  Boston,  Mar.  4,  1897. 
Sixty  days  from  date,  I  promise  to  pay  to  James  Finn,  or 

order,  Four  Thousand  Three  Hundred  Twenty-five  and  T%°<y 
Dollars,  value  received. 

Payable  at  the  Merchants  National  Bank. 

Discounted  at  5£%,  Mar.  8.  George  Bellows. 

3.  $1300.  Richmond,  Va.,  July  14,  1897. 
Ninety  days  from  date,  I  promise  to  pay  to  the  order  of 

Peter  Bright  Thirteen  Hundred  Dollars,  value  received. 
Payable  at  the  First  National  Bank. 
Discounted  at  4%,  Aug.  3.  George  Wright. 

4.  $1456.30.  Charleston,  S.  C,  Aug.  27,  1897. 
Three  months  after  date,  I  promise  to  pay  to  the  order  of 

John  George  Fourteen  Hundred  Fifty-six  and  T%%  Dollars, 
value  received. 

Payable  at  the  Second  National  Bank. 

Discounted  at  5%,  Sept.  10.  John  Waldorf. 

5.  $4550.36.  Baltimore,  Md.,  Nov.  10,  1897. 
Four  months  after  date,  I  promise  to  pay  to  the  order  of 

John  Callender  Four  Thousand  Five  Hundred  Fifty  and 
T3^  Dollars,  value  received. 

Payable  at  the  National  Mechanics  Bank. 

Discounted  at  5£%,  Nov.  24.  James  Barton. 


INTEREST   AND   DISCOUNT.  275 

6.  $5000.  Chicago,  III.,  Dec.  23,  1897. 
Six  months  after  date,  we  jointly  and  severally  promise 

to  pay  to  John  Adams,  or  order,  Five  Thousand  Dollars, 
value  received,  with  interest  at  five  per  cent. 

Payable  at  the  Metropolitan  National  Bank. 

Discounted  at  4%,  Jan.  21,  1898.        William  Dunn, 

F.  E.  Crockett. 

Find  the  day  of  maturity,  the  time  to  run,  the  discount, 
and  the  proceeds  of  the  following  notes,  with  grace  : 

7.  $4760.  Milwaukee,  Wis.,  Jan.  1,  1897. 
Ninety  days  after  date,  I  promise  to  pay  to  the  order  of 

James  Pike  Four  Thousand  Seven  Hundred  Sixty  Dollars, 

value  received. 

,    Payable  at  the  Wisconsin  National  Bank. 

Discounted  at  4^-%,  Feb.  15.  William  Clement. 

8.  $2017.85.  St.  Paul,  Minn.,  Jan.  14,  1897. 
Three  months  after  date,  I  promise  to  pay  to  the  order 

of  John  Brown  Two  Thousand  Seventeen  and  T8n5^  Dollars, 
value  received. 

Payable  at  the  German- American  National  Bank. 

Discounted  at  7%,  Mar.  1.  Timothy  Bruce. 

9.  $9040.  Galveston,  Tex.,  Jan.  19,  1897. 
Sixty  days  from  date,  I  promise  to  pay  to  the  order 

of  Charles  Carroll  Nine  Thousand  Forty  Dollars,  value 
received. 

Payable  at  the  First  National  Bank. 

Discounted  at  5£%,  Feb.  16.  James  Monroe. 

10.  $215.  Augusta,  Me.,  Jan.  28,  1897. 
Thirty  days  after  date,  I  promise  to  pay  to  the  order  of 

James  Fogg  Two  Hundred  Fifteen  Dollars,  value  received. 
Payable  at  the  Maine  National  Bank. 
Discounted  at  6%,  Feb.  3.  John  Moses. 


276  INTEREST   AND   DISCOUNT. 

11.  $2216.85.  Omaha,  Neb.,  Dec.  15,  1897. 
Ninety  days  after  date,  I  promise  to  pay  to  the  order  of 

F.  C.  Green  Two  Thousand  Two  Hundred  Sixteen  and  T8^ 
Dollars,  value  received. 

Payable  at  the  Omaha  National  Bank. 

Discounted  at  7%,  Jan.  8,  1898. 

W.  C.  Colburn. 

Find  the  proceeds  of  the  following  drafts,  with  grace  : 

12.  Draft  for  $620  at  60  days;  rate  of  discount  6%  ; 
exchange  £%. 

Solution.  The  discount  for  63  days  is  10£  x  §0.620,  or  $6.51 ;  and 
the  exchange  is  \%  of  $700,  or  $0.88  (every  fraction  of  $100  being 
reckoned  as  $100).  The  total  discount  is,  therefore,  $6.51  +  $0.88 
=  $7.39,  and  the  proceeds  is  $620  —  $7.39  =  $612.61. 

13.  Draft  for  $890  at  90  days  ;  rate  of  discount  4£%  ; 
exchange  £%. 

14.  Draft  for  $12,500  at  60  days  ;  rate  of  discount  5%  ; 
exchange  15  cents  on  $1000. 

15.  Draft  for  $1260  at  30  days  ;  rate  of  discount  5£%  ; 
exchange  \cf0. 

16.  Draft  for  $1430  at  3  months  ;  rate  of  discount  6%  ; 
exchange  \o/0. 

17.  Draft  for  $1875  at  4  months  ;  rate  of  discount  5%  ; 
exchange  \°f0. 

18.  Draft  for  $22,843  at  60  days  ;  rate  of  discount  4£% ; 
exchange  25  cents  on  $1000. 

19.  Draft  for  $18,000  at  2  months;  rate  of  discount 
5%  ;   exchange  •£%. 

20.  Draft  for  $3437.50  at  90  days  ;  rate  of  discount 
5%  ;   exchange  ±%. 

21.  Draft  for  $1287.50  at  60  days;  rate  of  discount 
4£%  ;    exchange  f  %. 

22.  Draft  for  $866.65  at  3  months  ;  rate  of  discount 
5%  ;    exchange  £%. 


INTEREST   AND   DISCOUNT.  277 

Present  Worth  and  True  Discount. 

487.  The  present  worth  of  a  debt  is  the  sum  which,  if 
put  at  interest,  will  amount  to  the  debt  when  due. 

488.  The  true  discount  is  the  difference  between  a  sum 
due  at  some  future  time  and  the  present  worth  of  that  sum. 

489.  Example.     Find  the  present  worth  and  true  dis- 
count of  $515  due  in  7  mo.  6  dy.,  if  money  is  worth  5%. 

Solution.     The  amount  of  $1  at  6%  for  7  mo.  6  dy.  is  $1.03. 

As  $1.03  is  the  amount  of  $1,  $515  is  the  amount  of  $j*.V|,  or  $500. 

The  true  discount  is  $515  —  $500  =  $15.     Hence, 

490.  To  Find  the  Present  Worth  of  a  Given  Sum  of 
Money  Due  at  a  Stated  Future  Time, 

Divide  the  given  sum  by  the  amount  of  $1  for  the  given 
rate  and  time. 

Exercise  125. 

1.  Find  the  present  worth  of  $500  due  in  11  mo.,  if 
money  is  worth  5°/0. 

2.  Find  the  present  worth  and  discount  of  $3334.62  due 
in  2  yr.,  if  money  is  worth  4-J-%. 

3.  Find  the  present  worth  and  discount  of  $4261.33  due 
in  l„yr.  6  mo.,  if  money  is  worth  6%. 

4.  Find  the  present  worth  and  discount  of  $2416.50  due 
in  7  mo.,  if  money  is  worth  5%. 

5.  Find  the  present  worth  of  $678.40  due  in  16  mo.,  if 
money  is  worth  k\°j0' 

6.  Find  the  present  worth  and  discount  of  $574.17  due 
in  2  yr.  3  mo.,  if  money  is  worth  h\°f0. 

7.  Find  the  present  worth  and  discount  of  $625.13  due 
in  8  mo.,  if  money  is  worth  4%. 

8.  Find  the  present  worth  and  discount  of  $715.20  due 
in  1  yr.  4  mo.,  if  money  is  worth  ?>\°J0. 


278  INTEREST   AND   DISCOUNT. 

Exact  Interest. 

491.  By  the  ordinary  method  interest  is  computed  on  a 
basis  of  30  days  for  a  month  ;  that  is,  360  days  for  a  year. 

A  year  has  365  days.  The  interest  for  a  year  counted 
by  days  would  thus  be  §  §#,  or  \\,  of  the  true  interest. 

The  exact  interest  for  any  number  of  days  is,  therefore, 
found  by  diminishing  interest  found  by  the  ordinary  method 
by  t1^  of  itself. 

492.  To  Find  One  Seventy-third  of  a  Number, 
Move  the  decimal  point  two  places  to  the  left,  add  a  third 

three  times,  each  time  one  place  further  to  the  right.     Carry 
only  to  three  decimal  places. 

493.  Example.  Find  the  exact  interest  on  $12,762.44 
from  May  6  to  Sept.  15,  at  6%. 

Solution.        26  $12.76244  $280.77 


30 

22 

$2.8077 

31 

2562488 

0.9369 

31 

2562488 

0.0935 

15 

1280.77368 

0.0093 

6|T32 

$3.8464 

22 

$276.92 

494.  Exact  interest  is  used  by  trust  companies,  and  in 
all  calculations  of  national  and  state  governments. 

Note.  This  rule  applies  only  to  periods  less  than  one  year.  For 
periods  greater  than  one  year,  we  find  the  interest  for  the  years,  then 
for  the  months  and  days  by  the  above  rule,  and  add  the  results. 

Exercise  126. 
Find  the  exact  interest  at  6%  : 

1.  On  $692.74  for  250  days. 

2.  On  $1472.38  from  Jan.  7,  1897  to  Oct.  4,  1897. 
8.    On  $1247.75  from  Mar.  4,  1897  to  Dec.  22,  1897. 
4.    On  $1898.48  from  Feb.  26,  1897  to  Aug.  12,  1899. 


INTEREST   AND   DISCOUNT.  279 


Annual  Interest. 

495.  Annual  interest  is  simple  interest  on  the  principal 
and  on  each  year's  interest  from  the  time  each  interest  is 
due  until  settlement. 

496.  Example.  Find  the  interest  on  $400  for  4  yr. 
7  mo.  20  dy.,  at  5%,  payable  annually. 

Solution.     Simple  interest  on  $400  at  5%  for  4  yr.  7  mo.  20  dy.  is 

$92.78. 

Interest  due  the  1st  year,  $20,  draws  interest  3  yr.  7  mo.  20  dy. 
Interest  due  the  2d  year,  $20,  draws  interest  2  yr.  7  mo.  20  dy. 
Interest  due  the  3d  year,  $20,  draws  interest  1  yr.  7  mo.  20  dy. 
Interest  due  the  4th  year,  $20,  draws  interest  7  mo.  20  dy. 

Interest  upon  the  interest  =  interest  on  $20  for   8  yr.  6  mo.  20  dy. 
Interest  on  $20  for  8  yr.  6  mo.  20  dy.  at  5%  =  $8.56. 
The  annual  interest  =  $92.78  +  $8.56  =  $101.34. 

Exercise  127. 

1.  Find  the  amount  at  annual  interest  of  $1247.75  for 
3  yr.  5  mo.  10  dy.  at  6%. 

2.  Find  the  interest  due  on  $987.25  in  4  yr.  9  mo.  6  dy., 
interest  at  4%,  payable  annually. 

3.  Find  the  interest  due  on  $742.60  in  5  yr.  11  mo.  27 
dy.,  interest  at  4£%,  payable  annually. 

4.  Find  the  interest  due  May  19,  1898,  on  a  note  dated 
Dec.  26, 1894,  for  $1224.60,  with  interest  payable  annually, 
at  5%,  if  no  interest  has  been  paid. 

5.  Find  the  amount  due  May  27,  1898,  on  a  note  dated 
Jan.  4,  1896,  for  $215.50,  with  interest  payable  annually 
at  5£%,  if  no  interest  has  been  paid. 

6.  Find  the  amount  due  Jan.  16, 1897,  on  a  note  dated 
Jan.  8,  1895,  for  $3115.20,  with  interest  payable  annually 
at  5%,  if  no  interest  has  been  paid. 


280  INTEREST   AND   DISCOUNT. 

Compound  Interest. 

497.  Interest  is  compounded  when  it  is  added  to,  and 
becomes  a  part  of,  the  principal  at  specified  intervals. 

Interest  is  compounded  annually,  semi-annually,  or  quar- 
terly, according  to  agreement.  Interest  is  understood  to  be 
compounded  annually  unless  otherwise  stated. 

498.  Example.  Find  the  amount  of  $500  at  compound 
interest  for  3  years  at  4%.     Find  the  compound  interest. 

Interest  for  1st  year  is  4%  of  §500       =  $20.  Amount  is  $520. 

Interest  for  2d  year  is  4%  of  $520       =  $20.80.     Amount  is  $540.80. 
Interest  for  3d  year  is  4%  of  $540.80  =  $21.63.     Amount  is  $562.43. 
The  compound  interest  is  the  compound  amount  less  the  principal ; 
that  is,  $562.43  —  $500  =  $62.43. 

Note.  If  the  time  is  not  an  integral  number  of  years,  the  com- 
pound amount  is  found  for  the  number  of  entire  years,  and  the 
amount  of  this  sum  at  simple  interest  for  the  portion  of  a  year. 

Exercise  128. 

1.  Find  the  amount  of  $356.25  for  4  yr.,  at  5%  com- 
pound interest. 

2.  Find  the  amount  of  $637.50  for  2  yr.  6  mo.,  at  4% 
compound  interest. 

3.  Find  the  compound  interest  on  $800  for  3  yr.  9  mo., 
at  6%. 

4.  Find  the  compound  interest  on  $39.35  for  4  yr.  9  mo., 
at  5%. 

5.  Find  the  compound  interest  on  $300  for  2  yr.,  at  4%, 
interest  being  compounded  semi-annually. 

Hint.     The  interest  is  2%  every  six  months. 

6.  Find  the  compound  interest  on  $525  for  1  yr.  6  mo., 
at  5%,  interest  being  compounded  quarterly. 

7.  Find  the  compound  interest  on  $10,000  for  6  mo.,  at 
6%,  interest  being  compounded  monthly. 


INTEREST    AND   DISCOUNT.  281 

Partial  Payments. 

499.  Partial  payments,  as  the  term  implies,  are  pay- 
ments of  a  part  of  a  note. 

There  are  several  methods  of  computing  interest  upon  such  notes, 
all  recognizing  the  general  principle  that  interest  should  cease  upon 
payments  made. 

500.  Merchants'  Rule.  If  a  note  that  bears  interest 
runs  one  year  or  less,  and  partial  payments  have  been  made, 
the  interest  is  computed  by  the  following  rule  : 

1.  Find  the  amount  of  the  note  at  date  of  settlement  with- 
out regarding  payments. 

2.  Find  the  amount  of  each  payment  with  interest  from 
date  of  payment  to  date  of  settlement. 

3.  Subtract  the  sum  of  the  payment  amounts  from  the  total 
amount. 

501.  Example.  A  man  holds  a  note  of  $460,  dated 
Jan.  20,  1897,  on  which  the  following  payments  are  en- 
dorsed :  $140,  Mar.  26,  1897;  $100,  June  16,  1897; 
$160,  Oct.  14,  1897. 

Settlement  is  made  Dec.  22,  1897.  Find  the  balance 
due,  reckoning  interest  at  6%. 

Solution. 

Time  from  Jan.  20  to  Dec.  22  is  11  mo.    2  dy.  Int.  on  $1  =  $0.055£. 

Time  from  Mar.  26  to  Dec.  22  is    8  mo.  26  dy.  Int.  on  $1  =    0.04% 

Time  from  June  16  to  Dec.  22  is    6  mo.    6  dy.  Int.  on  $1  —    0.031. 

Time  from  Oct.  14  to  Dec.  22  is    2  mo.    8  dy.  Int.  on  $1  ==    0.011£. 

Int.  on  $460  =  460  X  $0.055|  =  $25.45.      Amount  is  $485.45 

Int.  on  140  =  140  X  0.044^  =  6.21.  Amount  is  $146.21 
Int.  on  100=  100  X  0.031  =  3.10.  Amount  is  103.10 
Int.  on    160  =  160  X    0.011|  =      1-81.      Amount  is     161.81 

Total  payment  amounts  $411.12 

Balance  due  Dec.  22,  1897  $74.33 


282  INTEREST   AND   DISCOUNT. 

Exercise  129. 

1.  A  note  of  $618.75,  dated  Apr.  17,  1897,  payable  on 
demand,  bears  the  following  endorsements :  June  5, 
$126.50;  Aug.  20,  $137.25;  Nov.  17,  $210.  What  is 
due  Jan.  1,  1898,  reckoning  interest  at  6%  ? 

2.  A  note  of  $1000,  dated  Apr.  1,  1897,  payable  on 
demand,  with  interest  at  5%,  bears  the  following  endorse- 
ments :  May  6,  $200;  July  5,  $225.37;  Oct.  18,  $322. 
What  is  due  Jan.  1,  1898  ? 

3.  A  note  of  $835.25,  dated  July  1,  1897,  payable  on 
demand,  with  interest  at  4£%,  bears  the  following  endorse- 
ments :  Aug.  20,  $157.50  ;  Sept.  21,  $180.25 ;  Oct.  5, 
$200  ;   Dec.  1,  $80.     What  is  due  Jan.  1,  1898  ? 

4.  A  note  of  $1247.50,  dated  Mar.  10, 1897,  payable  on 
demand,  with  interest  at  5%,  has  the  following  endorse- 
ments :  $350.40,  Apr.  14,  1897  ;  $212.85,  June  16,  1897  ; 
$316.45,  Aug.  25,  1897.     What  is  due  Oct.  18,  1897  ? 

5.  A  note  of  $1648.25,  dated  Jan.  22, 1897,  payable  on 
demand,  with  interest  at  5%,  has  the  following  endorse- 
ments :  $212.60,  Mar.  1,  1897  ;  $168.40,  May  26,  1897  ; 
$244.40,  Aug.  4,  1897  ;  $744.80,  Oct.  1,  1897.  What  is 
due  Jan.  22,  1898  ? 

502.  United  States  Rule.  If  a  note  that  bears  interest 
runs  more  than  one  year,  and  partial  payments  have  been 
made,  the  interest  is  computed  by  the  following  rule  : 

1.  Find  the  amount  of  the  principal  to  the  time  when  the 
paymenb>  or  sum  of  the  payments,  is  equal  to  or  greater  than 
the  interest. 

2.  From  this  amount  deduct  the  payment,  or  sum  of  the 
payments. 

3.  Consider  the  remainder  as  a  new  principal,  and  pro- 
ceed as  before. 


INTEREST   AND   DISCOUNT.  288 

503.  Example.  A  note  of  $1520,  dated  May  20,  1896, 
bearing  interest  at  6%,  had  payments  endorsed  upon  it  as 
follows  :  Oct.  2,  1896,  $300 ;  Feb.  26,  1897,  $25  ;  Apr.  2, 
1897,  $570  ;  Aug.  9,  1897,  $600.  Find  the  amount  due 
Dec.  6,  1897. 

yr.       mo.     dy. 
1896     10      2  $1520  1st  principal. 

1896  5    20  0.022 

4     12      0.022  $33.44  1st  interest. 

1520.00 
$1553.44 

300.00  1st  payment. 

1897  2    26  $1253.44  2d  principal. 
1896     10       2  0.024 

4    24      0.024  $30.08  2d  interest. 


Since  the  interest  is  greater  than  the  payment,  we  find  the  interest 
on  the  same  principal  from  Feb.  26,  1897,  to  Apr.  2,  1897. 

$1253.44  2d  principal. 
0.006 
$7.52  3d  interest. 

1897      4      2  30.08  2d  interest. 

1897      2    26  .  1253.44 


1 

6 

0.006 

$1291.04 

$25  +  $570  =      595.00  2d  &  3d  payments. 
$696.04  3d  principal. 
0.02H 

1897 

8 

9 

$14.73  4th  interest. 

1897 

4 

2 

696.04 

0.021£  $710.77 

600.00 
$110.77  4th  principal. 
0.019^ 
1897     12      6  $2.16  5th  interest. 

1897       8      9  110.77 

3    27      0.019|  $112.93  due  Dec.  6,  1897. 


284  INTEREST    AND    DISCOUNT. 

Exercise  130. 

1.  A  note  of  $2000,  dated  Jan.  22,  1896,  and  drawing 
interest  at  6%,  had  the  following  endorsements  :  May  20, 
1896,  $100  ;  July  20,  1896,  $325  ;  Nov.  2,  1896,  $20  ; 
Dec.  23,  1896,  $125.     Find  the  balance  due  Mar.  1,  1897. 

2.  A  note  of  $1662.50,  dated  Jan.  15, 1896,  and  drawing 
interest  at  5£%,  had  the  following  endorsements  :    Apr.  30, 

1896,  $25  ;  June  24, 1896,  $25  ;  Sept.  2, 1896,  $625  ;  Jan. 
30,  1897,  $700.     Find  the  balance  due  May  12,  1897. 

3.  A  note  of  $4560,  dated  Jan.  22,  1896,  and  drawing 
interest  at  5%,  had  the  following  endorsements  :   Jan.  11, 

1897,  $2000  ;  Aug.  31,  1897,  $500 ;  Jan.  15,  1898,  $1200; 
Mar.  4,  1898,  $860.     Find  the  balance  due  June  15,  1898. 

4.  A  note  of  $785.50,  dated  Jan.  30, 1896,  and  drawing 
interest  at  5%,  had  the  following  endorsements :  July  17, 
1896,  $100;  Jan.  29,  1897,  $100;  Dec  31,  1897,  $20; 
Mar.  16,  1898,  $300  ;  June  18,  1898,  $50.  Find  the  bal- 
ance due  July  23,  1898. 

5.  A  note  of  $300.25,  dated  Aug.  4,  1896,  and  drawing 
interest  at  4£%,  had  the  following  endorsements  :  Oct.  14, 
1896,  $100;  July  21,  1897,  $100;  Oct.  11,  1897,  $50; 
Jan.  19,  1898,  $50.     Find  the  amount  due  July  22,  1898. 

6.  A  note  of  $1475.40,  dated  Feb.  12, 1896,  and  bearing 
interest  at  5%,  had  the  following  endorsements  :  July  22, 
1896,  $370;  Dec.  26,  1896,  $426.50;  Aug.  24,  1897, 
$112.40  ;  Oct.  6,  1897,  $163.25  ;  Apr.  14,  1898,  $185.85. 
Find  the  balance  due  June  16,  1899. 

7.  A  note  of  $5762.45  dated  Jan.  2, 1896,  and  drawing 
interest  at  5%,  had  the  following  endorsements  :  May  17, 
1896,  $500  ;  Oct.  12,  1896,  $750  ;  Feb.  4,  1897,  $1000  ; 
Aug.  25,  1897,  $1250;    Mar.  1,  1898,  $1500;    June  15, 

1898,  $1050.     Find  the  balance  due  Oct.  2,  1898. 


INTEREST   AND   DISCOUNT.  285 

Special  Rules  for  Partial  Payments. 
The  New  Hampshire  Rule. 

504.  When  a  note  is  written  with  interest  annually  or 
with  annual  interest  the  following  is  the  New  Hampshire 
Rule: 

If  in  any  year,  reckoning  from  the  time  the  annual  interest 
began  to  accrue,  payments  have  been  made,  compute  interest 
upon  them  to  the  end  of  the  year  in  which  they  are  made. 

The  amount  of  the  payments  is  to  be  then  applied,  first, 
to  cancel  interest  upon  annual  interest;  secondly,  to  cancel 
annual  interest ;  and  thirdly,  to  the  extinguishment  of  the 
principal. 

If,  however,  at  the  date  of  any  payment  there  is  no  interest 
except  the  accruing  annual  interest,  and  the  payment,  or  pay- 
ments, do  not  exceed  the  annual  interest  at  the  end  of  the  year, 
deduct  the  payment,  or  payments,  without  interest  on  the  same. 

The  Vermont  Rule. 

505.  If  we  omit  the  last  paragraph  of  the  New  Hamp- 
shire Rule,  we  have  the  Vermont  Rule. 

The  Connecticut  Rule. 

506.  The  Connecticut  Rule  for  partial  payments  is  : 
Follow  the  United  States  Mule  if  a  year's  interest  or  more 

is  due  at  the  time  of  a  payment,  and  in  case  of  the  last  pay- 
ment. 

If  less  than  a  year's  interest  is  due  at  the  time  of  a  pay- 
ment except  the  last,  find  for  a  new  principal  the  difference 
between  the  amount  of  the  principal  for  an  entire  year  and 
the  amount  of  the  payment  for  the  remainder  of  that  year. 

If  the  interest  due  at  the  time  of  a  payment  exceeds  the 
payment,  find  the  interest  on  the  principal  only. 


286  INTEREST   AND   DISCOUNT. 

507.  Examples.  1.  Find,  by  the  New  Hampshire  Rule, 
the  amount  due  Sept.  22,  1896,  on  a  note  for  $500,  dated 
July  16,  1893,  with  interest  annually  at  6%,  on  which  the 
following  payments  have  been  made  :  Oct.  16, 1894,  $200  ; 
Dec.  16,  1895,  $20. 

Principal,  $500.00 

1st  annual  interest,  $30.00 

Int.  on  1st  annual  int.  for  1  yr.,  $1.80 

2d  annual  interest,  30.00 

$600.00  $60.00    $1.80 
Payment  Oct.  16,  1894,  |200 

Int.  on  payment  July  16,  1895,      9 

Amt.  of  payment  July  16,  1895,  $209  =  147.20  +  60.00  +  1.80 

Principal  July  16,  1895,  $352.80 

3d  annual  interest,  $21.17 

Payment  Dec.  16,  1895,  20.00 

$352.80  +  $1.17 

Principal  July  16,  1896,  $353.97 

4th  annual  interest,  3.89 

Amt.  due  Sept.  22,  1896,  $357.86 

2.  Find,  by  the  Vermont  Rule,  the  amount  due  Sept.  22, 
1896,  of  the  note  in  Example  1. 

From  Example  1,  the  principal  July  16,  1895,  is  $352.80. 

Principal  July  16,  1895,  $352.80 

3d  annual  interest,  $21.17 


Payment  Dec.  16,  1895, 

$20.00 

Int.  on  payment  July  16,  1896, 

0.70 

Amt.  of  payment  July  16,  1896, 

$20.70  = 

20.70 

$352.80+    $0.47 

Principal  July  16,  1896, 

$353.27 

4th  annual  interest, 

3.89 

Amt.  due  Sept.  22,  1896, 

$357.16 

3.  Find,  by  the  Connecticut  Rule,  the  amount  due  Sept. 
22,  1896,  on  a  note  for  $500  given  Apr.  10,  1892,  bearing 
interest  at  6%,  which  has  the  following  endorsements  : 


INTEREST   AND   DISCOUNT.  28T 

June  16, 1893,  $100 ;  Dec.  28, 1893,  $100s  May  10, 1895, 
$  15  ;  Mar.  4,  1896,  $200. 

Principal,  $500.00 

Int.  on  principal  to  June  16,  1893,  35.50 

Amt.  of  principal  June  16,  1893,  $535.50 

Payment  June  16,  1893,  100.00 

New  principal  June  16,  1893,  $435.50 

Int.  on  principal  to  June  16,  1894,  26.13 

Amt.  of  principal  June  16,  1894,  $461.63 

Payment  Dec.  28,  1893,  $100.00 

Int.  on  payment  to  June  16,  1894,  2.80 

Amt.  of  payment  June  16,  1894,  $102.80 

New  principal  June  16,  1894,  $358.83 

Int.  on  principal  to  June  16,  1895,  21.53 

Amt.  of  principal  June  16,  1895,  $380.36 

Payment  May  10,  1895  (less  than  interest),  15.00 

New  principal  June  16,  1895,  $365.36 

Int.  on  principal  to  Mar.  4,  1896,  15.71 

Amt.  of  principal  Mar.  4,  1896,  $381.07 

Payment  Mar.  4,  1896,  200.00 

New  principal  Mar.  4,  1896,  $181.07 

Int.  on  principal  to  Sept.  22,  1896,  5.98 

Amt.  due  Sept.  22,  1896,  $187.05 


Exercise  131. 

t-  Find,  by  the  New  Hampshire  Eule  and  also  by  the 
Vermont  Rule,  the  amount  due  Sept.  22,  1896,  on  a  note 
for  $1750,  dated  June  6,  1892,  with  interest  annually  at 
6%,  which  has  the  following  endorsements  :  Aug.  12,  1893, 
$300  ;  Dec.  23,  1893,  $200  ;  Jan.  15,  1895,  $50  ;  Apr.  23, 
1896,  $800. 

2.  Find,  by  the  Connecticut  Eule,  the  amount  due  Sept. 
22,  1896,  on  a  note  for  $1500,  dated  Aug.  9,  1892,  with 
interest  annually  at  6%,  which  has  the  following  endorse- 
ments :  Mar.. 17,  1893,  $250;  Apr.  19,  1894,  $50;  Sept. 
21,  1895,  $500  ;    June  26,  1896,  $600. 


CHAPTER  XIV. 

STOCKS  AND  BONDS,  EXCHANGE,  ACCOUNTS. 
Stocks  and  Bonds. 

508.  Stock  Companies.  A  stock  company  is  an  associa- 
tion of  persons  under  the  laws  of  the  state  for  the  purpose 
of  carrying  on  a  specified  business. 

509.  Stock.  The  stock  represents  the  capital  invested 
in  the  business,  and  is  issued  in  the  form  of  certip'mfrs, 
each  certifying  that  the  person  named  in  the  certificate 
owns  a  stated  number  of  shares  of  stock. 

Note.  One  of  the  special  advantages  of  a  number  of  persons  doing 
business  in  the  form  of  a  company  is  that  the  liability  of  each  stock- 
holder is  usually  limited  to  the  amount  of  his  stock. 

510.  Bonds  are  written  obligations  under  seal  given  by 
a  company  ;  or  by  a  municipal  or  state  government;  or  by 
the  national  government ;  in  which  an  agreement  is  made 
to  pay  a  specified  amount  on  or  before  a  specified  date,  with 
interest  payable  annually,  semi-annually,  or  quarterly. 

511.  Mortgage  bonds  are  bonds  secured  by  mortgage, 
but  debenture  bonds  are  simply  notes  under  seal. 

512.  Registered  bonds  are  bonds  recorded  by  their  num- 
bers in  a  book  called  a  register.  The  register  contains  the 
names  of  the  owners  of  the  bonds. 

Note.  Registered  bonds  are  transferred  from  one  person  to 
another  by  giving  proper  notice  to  the  registrar  of  the  company. 


STOCKS   AND   BONDS.  289 

513.  Coupon  bonds  are  bonds  that  have  coupons  or  cer- 
tificates of  interest  attached.  These  coupons  are  cut  off 
as  they  become  due,  and  are  given  up  on  receipt  of  the 
interest  represented  by  them. 

514.  Bonds  are  named  by  giving  the  name  of  the  cor- 
poration issuing  them,  and  the  rate  of  interest  they  bear. 
The  date  on  which  they  become  due  is  usually  given,  and 
the  kind  of  bond,  registered  or  coupon. 

Thus,  U.  S.  coupon  4's,  1907  means  United  States  coupon  bonds 
bearing  4  per  cent  interest,  the  principal  due  in  1907.  U.  S.  registered 
4's,  1925  means  United  States  registered  bonds  bearing  4%  interest,  the 
principal  due  in  1925. 

515.  Persons  who  buy  and  sell  stocks  and  bonds  are 
called  stock  brokers,  and  their  commission  is  called  broker- 
age. On  lots  of  100  shares  or  more,  brokerage  is  -J%  per 
share,  reckoned  at  $100;  and  on  lots  of  less  than  100 
shares  brokerage  is  \°]0  per  share.  Brokerage  on  bonds  is 
reckoned  at  the  same  rate  per  $100  as  on  stocks. 

516.  Stocks  and  bonds  are  said  to  be  at  par  when  they 
sell  for  their  face  value  ; 

At  a  premium,  or  above  par,  when  they  sell  for  more 
than  their  face  value  ; 

At  a  discount,  or  below  par,  when  they  sell  for  less  than 
their  face  value. 

The  price  for  which  stocks  and  bonds  may  be  sold  is 
called  their  market  value. 

517.  A  company  uses  its  receipts  first  to  pay  its  ex- 
penses ;  second,  to  pay  the  interest  on  its  bonds.  If  there 
is  a  surplus,  it  is  usually  divided  among  the  stockholders, 
according  to  the  number  of  shares  held  by  each.  The 
amount  allotted  to  each  share  is  called  the  dividend  per 
share. 


290  STOCKS   AND   BONDS. 

Exercise  132. 

1.  What  is  the  cost  of  25  shares  of  Boston  and  Maine 
R.R.  stock  at  167,  brokerage  J  ? 

Solution.  Each  share  of  stock  costs  §167,  with  $0.25  for  bro- 
kerage, or  $167.26  in  all.  Hence,  26  shares  cost  26  X  $167.25,  or 
$4181.25. 

2.  How  many  shares  of  Illinois  Central  R.R.  stock  at 
101J  can  be  bought  for  $20,400,  brokerage  i  ? 

Solution.  Since  the  total  cost  of  each  share  is  $101$  +  $$-,  or 
$102,  $20,400  will  buy  *£#§fi  shares,  or  200  shares. 

3.  What  is  the  annual  income  from  150  shares  of  Lake 
Shore  and  Michigan  Ry.  stock  that  pays  an  annual  divi- 
dend of  6%  ? 

Solution.  1  share  at  6%  yields  $6.  Hence,  160  shares  at  6%  yield 
160  X  $6  =  $900. 

4.  How  much  must  be  invested  in  6%  stock  at  107  to 
yield  an  annual  income  of  $240,  brokerage  £  ? 

Solution.        $6  is  the  dividend  from  1  share. 

$240  dividend  requires  2Jft,  or  40  shares. 
Price  of  40  shares  is  40  X  $107  =  $4280 

Brokerage  on  40  shares  is  40  X  $0.25  = 10 

Total  cost  =  $4290 

5.  What  per  cent  does  the  investment  yield,  if  Lake 
Shore  and  Michigan  Southern  Ry.  stock  is  bought  at  170  ? 
The  stock  pays  6%  dividend  ;    no  brokerage  reckoned. 

Solution.  Each  $170  invested  yields  $6  dividend.  Hence,  the 
income  on  the  investment  is  Tf  ^  of  100%,  or  3^7%. 

6.  Find  the  cost  of  350  shares  of  Chicago,  Milwaukee 
and  St.  Paul  Ry.  stock  at  93f ,  brokerage  £. 

7.  Find  the  cost  of  165  shares  of  Michigan  Central  R.R. 
stock  at  105f ,  brokerage  £. 

8.  Find  the  cost  of  35  shares  of  Reading  R.R.  stock  at 
23f ,  brokerage  £. 


STOCKS    AND    BONDS.  291 

9.    What  is  the  cost  of  25  U.  S.  4%  registered  1925 
bonds  of  $1000  each,  at  127-J,  brokerage  £? 

10.  What  is  the  cost  of  40  Northern  Pacific  R.R.  1st 
mortgage  6%  registered  bonds  of  $1000  each,  at  119 J,  bro- 
kerage -J  ? 

11.  What  per  cent  income  does  the  investment  of  Ex- 
ample 10  yield  ? 

12.  What  is  the  annual  income  received  from  the  invest- 
ment of  Example  10  ? 

13.  What  is  the  annual  income  from  200  shares  of 
Chicago  and  Northwestern  Ry.  stock  that  pays  an  annual 
dividend  of  5%  ? 

14.  What  is  the  cost  of  the  investment  of  Example  13 
at  122J,  brokerage  £  ? 

15.  What  per  cent  income  does  the  investment  of  Ex- 
ample 13  yield  ? 

16.  How  many  shares  of  New  York  Central  stock  can 
be  bought  for  $4757.50  at  107J-,  brokerage  £  ? 

17.  How  many  Chicago,  Burlington  and  Quincy  7% 
bonds,  $500  each,  will  $6885  buy  at  114£,  brokerage  J  ? 

18.  What  is  the  annual  income  from  the  investment  of 
Example  17  ? 

19.  What  sum  of  money  must  be  invested  in  Northern 
Pacific  R.R.  1st  mortgage  6's  at  119£  to  produce  an  annual 
income  of  $2400,  brokerage  -J-  ? 

20.  What  sum  of  money  must  be  invested  in  Wabash 
R.R.  1st  gold  5%  bonds  at  107J  to  produce  an  annual 
income  of  $2000,  brokerage  -J-  ? 

21.  What  sum  of  money  must  be  invested  in  Louisville 
and  Nashville  R.R.  unified  gold  4%  bonds  at  84£  to  produce 
an  annual  income  of  $320,  brokerage  \  ? 

22.  What  sum  of  money  must  be  invested  in  St.  Louis 
and  San  Francisco  Ry.  general  mortgage  5  °J0  bonds  at  100 \ 
to  produce  an  annual  income  of  $600,  brokerage  -J  ? 


292  STOCKS   AND   BONDS. 

23.  How  many  shares  of  Chicago  and  Northwestern  Ry. 
stock  can  be.  bought  for  $14,670  at  122£,  brokerage  £  ? 
What  is  the  brokerage?  If  5%  dividends  are  paid,  what 
per  cent  on  his  investment  does  the  purchaser  receive  ? 

24.  How  many  shares  of  Michigan  Central  R.R.  stock 
can  be  bought  for  $16,940  at  105f,  brokerage  i  ?  What  is 
the  brokerage  ?  If  4%  dividends  are  paid,  what  per  cent 
on  his  investment  does  the  purchaser  receive  ? 

25.  What  is  the  cost  of  40  shares  of  Central  R.R.  of  New 
Jersey  stock  at  92J,  brokerage  J  ?  What  is  the  brokerage  ? 
If  6%  dividends  are  paid,  what  per  cent  on  his  investment 
does  the  purchaser  receive  ? 

26.  What  is  the  cost  of  250  shares  of  Pullman  Palace 
Car  Co.  stock  at  171J,  brokerage  £  ?  What  is  the  broker- 
age ?  If  8%  dividends  are  paid,  what  per  cent  on  his 
investment  does  the  purchaser  receive  ? 

27.  What  per  cent  on  his  investment  does  a  purchaser 
receive  who  buys  New  York,  New  Haven  and  Hartford 
R.R.  stock  at  180£,  if  annual  dividends  of  8%  are 
declared  ? 

28.  When  West  End  Co.  4£%  bonds  are  selling  at  107£, 
how  much  must  be  invested  to  produce  an  annual  income 
of  $900,  brokerage  £  ?  What  per  cent  on  his  investment 
does  a  purchaser  receive  ? 

29.  When  Mexican  Central  Ry.  1st  mortgage  4%  bonds 
are  selling  at  62£,  how  much  must  be  invested  to  produce 
an  annual  income  of  $200,  brokerage  £  ?  What  per  cent 
on  his  investment  does  a  purchaser  receive  ? 

30.  When  West  Shore  R.R.  1st  mortgage  4%  bonds  are 
selling  at  108£,  how  much  must  be  invested  to  produce  an 
annual  income  of  $800,  brokerage  £  ? 

31.  When  New  England  Tel.  and  Tel.  Co.  6%  bonds  are 
selling  at  101£,  how  much  must  be  invested  to  produce  an 
annual  income  of  $900,  brokerage  -J  ? 


STOCKS   AND   BONDS.  293 

32.  If  a  man  buys  a  6%  bond  at  120,  what  rate  of 
interest  does  he  receive  on  the  money  invested  ? 

33.  If  3%  bonds  are  at  88-J-,  what  rate  per  cent  interest 
will  a  purchaser  receive  on  his  money  ? 

34.  If  an  8%  stock  is  at  150,  what  rate  per  cent  interest 
will  a  purchaser  receive  on  his  money  ? 

35.  If  a  10%  stock  is  at  175,  what  rate  per  cent  interest 
will  an  investor  receive  on  his  money  ? 

36.  If  a  4£%  stock  is  at  85,  what  rate  per  cent  interest 
will  a  purchaser  receive  on  his  money  ? 

37.  If  7%  bonds  are  at  114,  what  rate  per  cent  interest 
will  a  purchaser  receive  on  his  money  ? 

38.  If  6%  bonds  are  at  130,  what  rate  per  cent  interest 
will  a  purchaser  receive  on  his  money  ? 

39.  If  $8000  5%  stocks  are  sold  at  90,  and  the  proceeds 
invested  in  3^-%  stocks  at  60,  find  the  increase  or  decrease 
in  income. 

40.  If  $10,000  3£%  bonds  are  sold  at  65,  and  the  pro- 
ceeds invested  in  8%  bonds  at  130,  find  the  increase  or 
decrease  in  income. 

41.  If  $8000  4£%  stocks  are  sold  at  70,  and  the  pro- 
ceeds invested  in  10%  stocks  at  160,  find  the  increase  or 
decrease  in  income. 

42.  If  $6000  6%  bonds  are  sold  at  90,  and  the  pro- 
ceeds invested  in  10%  bonds  at  135,  find  the  increase  or 
decrease  in  income. 

43.  Find  the  rate  of  interest  obtained  by  investing  in  a 
5%  bond  at  124. 

44.  What  is  the  price  of  stock  if  $7000  stock  can  be 
bought  for  $5880  ? 

45.  Find  the  amount  received  for  100  mining  shares 
issued  at  $15  a  share  and  sold  at  2\°]0  discount. 

46.  How  much  3£%  stock  must  be  sold  at  75 \  to  buy 
$5000  4%  stock  at  94|,  brokerage  \  on  each  transaction? 


294  STOCKS   AND   BONDS. 

47.  How  much  stock  must  be  sold  at  76^  to  raise  a  sum 
sufficient  to  discount  a  note  for  $1075,  due  in  53  days,  with 
grace,  and  discounted  at  5-J-%  ? 

48.  A  broker  bought  five  $1000  bonds  at  88|.  At  what 
price  must  he  sell  them  to  gain  $100,  brokerage  £  on  each 
transaction  ? 

49.  If  a  broker  buys  bonds  at  87 J,  at  what  price  must 
he  sell  them  to  make  12£%  profit,  brokerage  £  on  each 
transaction  ? 

50.  Which  is  the  more  profitable  stock  for  investment, 
a  4%  at  85  or  a  3%  at  63  ?   a  3J%  at  67J  or  a  4%  at  81£  ? 

51.  Find  the  price  of  a  4£%  bond  to  be  as  profitable  an 
investment  as  a  3£%  bond  at  88£. 

52.  Find  the  price  of  a  5%  bond  to  be  as  profitable  an 
investment  as  a  3%  bond  at  89 J. 

53.  Find  the  price  of  a  3£%  bond  to  be  as  profitable  an 
investment  as  a  6%  bond  at  par. 

54.  Find  the  loss  in  buying  $80,000  worth  of  bonds  at 
91f  and  selling  at  90,  brokerage  £  on  each  transaction. 

55.  Which  is  the  better  investment,  a  5%  stock  at  137 J- 
or  a  3£%  stock  at  91£?  What  rate  of  interest  will  be 
received  from  each  investment  ? 

56.  A  person  invests  $7370  in  the  purchase  of  a  stock 
at  92.  What  will  be  his  loss  if  he  sells  at  90,  brokerage  £ 
on  each  transaction  ? 

57.  How  much  stock  must  be  sold  at  90f  so  that  when 
the  seller  invests  the  proceeds  in  a  mortgage  at  6%  he 
will  receive  $543.75  annual  income  ? 

58.  A  person  invests  }  of  his  money  at  6%,  §  at  4£%, 
and  the  rest  at  3^%.  What  per  cent  does  he  receive  on 
the  whole  amount  ? 

59.  How  many  shares  of  stock  must  a  man  sell  at  107|, 
that  when  he  invests  the  proceeds  in  3%  stock  at  7l£  he 
may  receive  an  annual  income  of  $900  ? 


EXCHANGE.  295 

Exchange. 

518.  Exchange.  The  system  of  paying  debts  due  per- 
sons living  at  a  distance  by  transmitting  drafts  instead  of 
money  is  called  exchange. 

519.  Exchange  between  cities  of  the  same  country  is 
called  Domestic  Exchange.  Exchange  between  cities  of 
different  countries  is  called  Foreign  Exchange. 

520.  Examples.  1 .  Find  the  cost  of  the  following  draft 
with  grace,  interest  6%,  exchange  %°/0  premium  : 

$800.  Cincinnati,  O.,  Nov.  1,  1897. 

Thirty  days  after  sight,  pay  to  the  order  of  S.  G.  Clark 
Eight  Hundred  Dollars,  and  charge  to  the  account  of 

To  H.  S.  Wright,  Philadelphia.       P.  M.  Clement. 

Solution.     The  discount  on  $800  at  6%  for  33  days  is  $4.40. 
The  proceeds  of  the  draft  is  $800  —  $4.40  =  $795.60. 
The  exchange  is  \%  of  $800  =  $4. 
Hence,  the  cost  of  the  draft  is  $795.60  +  $4  =  $799.60. 

2.  Find  the  cost  of  a  draft  of  $400,  payable  60  days  after 
sight  with  grace,  interest  7%,  exchange  \°f0  discount. 

Solution.     The  discount  on  $400  at  7%  for  63  days  is  $4.90. 

The  exchange  is  ±%  of  $400  =  $1. 

The  total  discount  is  $4.90  +  $1  =  $5.90. 

Hence,  the  cost  of  the  draft  is  $400  —  $5.90  =  $394.10. 

3.  Find  the  face  of  a  draft,  payable  30  days  after  sight 
with  grace,  that  can  be  bought  for  $1000,  interest  6%, 
exchange  \°]0  premium. 

Solution.  The  discount  on  $1  for  33  days  at  6%  is  $0.0055  ;  and 
the  proceeds  of  $1  is  $1  —  $0.0055  =  $0.9945. 

The  exchange  on  $1  is  $0.0025  ;  and  the  cost  of  $1  is  $0.9945  + 
$0.0025  =  $0,997. 

Hence,  the  face  of  the  draft  is  $1000  -i-  0.997  =  $1003.01. 


296  EXCHANGE. 


Exercise  133. 


1.  Find  the  cost  of  a  sight  draft  on  New  York  of  $1100, 
exchange  ±%  premium. 

2.  Find  the  cost  of  a  sight  draft  on  New  Orleans  of 
$1350,  exchange  £%  discount. 

3.  Find  the  cost  of  a  draft  on  Boston  of  $1600,  payable 
30  days  after  sight  with  grace,  interest  6%,  exchange  \of0 
premium. 

4.  Find  the  cost  of  a  draft  of  $500,  payable  60  days 
after  sight  with  grace,  interest  7%,  exchange  £%  discount. 

5.  Find  the  cost  of  a  draft  of  $1200,  payable  90  days 
after  sight  with  grace,  interest  7%,  exchange  \°J0  premium. 

6.  Find  the  cost  of  a  draft  of  $950,  payable  in  30  days 
with  grace,  interest  4£%,  exchange  at  par. 

7.  Find  the  cost  of  a  draft  of  $725,  payable  in  60  days 
with  grace,  interest  5%,  exchange  \°J0  discount. 

8.  Find  the  cost  of  a  draft  of  $810,  payable  in  90  days 
with  grace,  interest  5£%,  exchange  J%  premium. 

9.  Find  the  face  of  a  draft,  payable  30  days  after  sight 
with  grace,  that  can  be  bought  for  $274,  interest  6%, 
exchange  at  par. 

10.  Find  the  face  of  a  draft,  payable  60  days  after  sight 
with  grace,  that  can  be  bought  for  $1250,  interest  7%, 
exchange  \°f0  premium. 

11.  Find  the  face  of  a  draft,  payable  60  days  after  date 
witli  grace,  that  can  be  bought  for  $1125,  interest  5£%, 
exchange  \°/0  discount. 

12.  Find  the  face  of  a  draft,  payable  30  days  after  date 
with  grace,  that  can  be  bought  for  $520,  interest  4%, 
exchange  \°j0  premium. 

13.  Find  the  face  of  a  draft,  payable  90  days  after  date 
with  grace,  that  can  be  bought  for  $10,000,  interest  4£%, 
exchange  at  par. 


EXCHANGE.  297 

521.  Foreign  Exchange.  Foreign  Bills  of  Exchange 
are  usually  drawn  in  sets  of  three,  of  the  same  tenor  and 
date,  called  First,  Second,  and  Third  of  Exchange.  These 
are  sent  by  different  mails  to  avoid  loss  or  delay.  When 
one  is  accepted  or  paid  the  others  are  void.  Thus, 
£600.  New  York,  Nov.  1,  1897. 

At  Sight  of  this  First  of  Exchange  (Second  and  Third  of 
the  same  tenor  and  date  unpaid),  pay  to  the  order  of  James 
Patterson  Five  Hundred  Pounds,  value  received,  and  charge 
same  to  account  of 

To  Baring  Blathers,     ^1  _,,  _  - 

T  _     ,      _    >  Bliss  and  Morton. 

London,  England.  J 

522.  Exchange  for  sight  drafts  Nov.  13,  1897,  was 
quoted  in  New  York  as  follows  : 

On  London  at  $4,865  for  1  pound  sterling. 

On  Paris  at  5.18£  francs  for  $1. 

On  cities  in  Germany  at  4  reichsmarks  for  $0,955. 

On  Amsterdam  at  1  guilder  for  $0.40£. 

523.  Examples.  1.  Find  the  cost  in  New  York  of  a 
sight  draft  on  London  for  £502  12s. 

Solution.  £502  12s.  =  £502.6. 

502.6  X  $4,865  =  $2445.15. 

2.  Find  the  face  of  a  sight  draft  on  London  that  can  be 
bought  for  $2433.47. 

Solution.     $2433.47  -f  $4,865  =  500.2. 

£500.2  =  £500  4s. 

3.  Find  the  cost  in  New  York  of  a  sight  draft  on  Paris 

for  2400  francs. 

1 


Solution.  1  franc  =  $ 


6.18* 


2400  francs  =  2400  X  $  -^—  =  $463.21. 
5.18$ 


298  EXCHANGE. 

Exercise  134. 
Find  the  cost  of  a  sight  draft  on  : 

1.  London  for  £320  10s.  6d. 

2.  Paris  for  8000  francs. 

3.  Hamburg  for  2876  reichsmarks. 

4.  Amsterdam  for  6486  guilders. 

5.  Glasgow  for  £5876  10s. 

6.  Paris  for  12,842  francs. 

7.  Berlin  for  4885  reichsmarks. 

8.  Kotterdam  for  8282  guilders. 

9.  Liverpool  for  £1242  12s.  6d. 

10.  Paris  for  2685  francs. 

1 1 .  Find  the  face  of  a  sight  draft  on  Glasgow  that  can 
be  bought  for  $2000. 

12.  Find  the  face  of  a  sight  draft  on  London  that  can 
be  bought  for  $4000. 

13.  Find  the  cost  of  a  sixty-day  draft  on  London  for 
£150,  when  sixty-day  bills  are  quoted  at  4.81£,  and  the 
broker's  commission  is  -J%  of  the  cost  of  the  draft. 

14.  How  large  a  sight  draft  on  Paris  can  be  bought  for 
$2840  ? 

15.  How  large  a  sixty-day  draft  on  Paris  can  be  bought 
for  $1500,  when  sixty-day  bills  are  quoted  at  5.17J  ? 

16.  How  large  a  sight  draft  on  Berlin  can  be  bought  for 
$8000  ? 

17.  How  large  a  sixty-day  draft  on  Hamburg  can  be 
bought  for  $2500,  when  German  sixty-day  drafts  are 
quoted  at  0.95  ? 

18.  How  large  a  sight  draft  on  Amsterdam  can  be  bought 
for  $2200  ? 

19.  How  large  a  sixty-day  draft  on  Kotterdam  can  be 
bought  for  $1200,  when  a  sixty-day  draft  on  Holland  is 
quoted  at  0.40|  ? 


AVERAGE   OF   PAYMENTS.  299 


Average  of  Payments. 

524.  Example.  John  Smith  has  given  to  William 
Jones  notes  as  follows  :  $150,  due  May  15  ;  $200,  due 
June  30  ;  $500,  due  July  21  ;  $400  due  July  29.  He 
wishes  to  pay  them  all  at  one  time.  When  shall  they  be 
considered  due  ? 

Solution.  If  all  the  notes  were  paid  May  15,  Smith  would  lose  the 
use  of  $200  for  46  days,  of  $500  for  67  days,  and  of  $400  for  75  days. 

The  use  of  $200  for  46  days  =  the  use  of  $200  X  46  for  1  day ;  the 
use  of  $500  for  67  days  =  the  use  of  $500  X  67  for  1  day ;  and  the 
use  of  $400  for  75  days  =  the  use  of  $400  X  75  for  1  day.  Smith 
would,  therefore,  lose  the  equivalent  of  $9200  +  $33,500  +  $30,000 
=  $72,700  for  1  day,  and  is  entitled  to  keep  the  $150  +  $200  +  $500 
+  $400  —  $1250  as  many  days  after  May  15  as  are  required  for  the 
use  of  $1250  to  equal  the  use  of  $72,700  for  1  day;  that  is,  VzV<r 
days  =  58. 2  days. 

Hence,  the  equated  time  for  paying  the  four  notes  is  58  days  after 
May  15 ;   that  is,  July  12. 

The  work  may  be  arranged  as  follows : 


$150  X  00 : 

$200  X  46  : 

=  $9,200 

$500  X  67  = 

=    33,500 

$400  X  75  = 

=  30,000 

$1250 

)$72,700 

58.2. 

Hence, 


525.  To  Find  the  Equated  Time  for  the  Payment  of 
Several  Debts  Due  at  Different  Dates, 

Select  the  earliest  date  that  any  debt  becomes  due  as  the 
standard  date. 

Multiply  each  of  the  debts  by  the  number  of  days  from 
the  standard  date  to  the  date  that  it  becomes  due,  and  divide 
the  sum,  of  the  products  by  the  sum  of  the  debts. 

The  quotient  is  the  number  of  days  that  must  be  added  to 
the  standard  date  to  find  the  average  time  of  the  payments. 


300  AVERAGE   OF   PAYMENTS. 

Exercise  135. 

1 .  Find  the  equated  time  for  the  payment  of  $250  due 
in  3  mo.,  $ 400  due  in  6  mo.,  $700  due  in  8  mo. 

2.  Find  the  equated  time  for  the  payment  of  $300  due 
in  30  days,  $500  due  in  60  days,  and  $200  due  in  90  days. 

3.  Find  the  equated  time  for  the  payment  of  $325  due 
now,  $200  due  in  30  days,  $460  due  in  60  days,  and  $150 
due  in  90  days. 

4.  Find  the  equated  time  for  the  payment  of  $240  due 
May  10,  $420  due  July  2,  $310  due  Sept.  14,  and  $600  due 
Oct.  1. 

5.  Find  the  equated  time  for  the  payment  of  $275  due 
June  21,  $175  due  July  16,  $200  due  Aug.  6,  and  $150 
due  Sept.  3. 

6.  Find  the  equated  time  for  the  payment  of  $112.30 
due  July  6,  $115.25  due  July  30,  $232.15  due  Sept.  4,  and 
$102.36  due  Oct.  1. 

7.  A  owes  B  $200  due  in  10  mo.  If  he  pays  $120  in 
4  mo.,  when  should  he  pay  the  balance  ? 

Note.  By  paying  3120  in  4  mo.  A  loses  the  use  of  $120  for  6  mo., 
which  is  equal  to  the  use  of  $720  for  1  mo.  Therefore,  he  is  entitled 
to  keep  the  balance  ($80)  \\°-  mo.  =  9  mo.  after  its  maturity. 

8.  A  owed  B  $2000  payable  in  4  mo.,  but  at  the  end  of 
1  mo.  he  paid  him  $500,  at  the  end  of  2  mo.  $500,  and 
at  the  end  of  3  mo.  $500.  In  how  many  months  is  the 
balance  due  ? 

9.  A  man,  Feb.  11,  1898,  gave  a  note  for  $1700  pay- 
able in  4  mo.;  but  he  paid  Mar.  22,  $400,  Apr.  20,  $220, 
May  10,  $300.     When  was  the  balance  due  ? 

1 0.  A  man,  Jan.  4,  1898,  gave  a  note  for  $2500  payable 
in  6  mo.;  but  he  paid  Feb.  4,  $200,  Mar.  4,  $400,  Apr.  4, 
$600,  May  4,  $500,  and  June  4,  $300.  When  was  the 
balance  due  ? 


SETTLEMENT    OF    ACCOUNTS. 


301 


Settlement  of  Accounts. 


526.    Examples.     1.    Find  the  time  for  the  payment  of 
the  balance  of  the  following  account : 


Adams  &  Co.  in  account  with  Bacon  &  Co. 


Dr. 


Cr. 


1898. 

Jan.  3. 

To  mdse.  90  dy. 

$250 

Apr.  11. 

By  cash, 

$200 

Mar.  7. 

60  " 

150 

Apr.  30. 

ft 

100 

May  3. 

60  " 

325 

May  30. 

M 

125 

June  7. 

"        30   " 

175 

July  2. 

u 

400 

Solution.  We  first  find  the  equated  time  of  the  items  on  the 
Debit  side  to  be  May  29,  1898 ;  and  the  equated  time  of  the  items  on 
the  Credit  side  to  be  May  30,  1898. 

The  total  of  the  Debit  side  is  $900,  and  the  total  of  the  Credit  side 
is  $825.     Hence,  the  balance  is  $900  —  $825  =  $75. 

The  difference  between  the  equated  times  is  1  day. 

If  the  account  were  settled  at  the  later  date,  May  30,  the  $900  on 
the  Debit  side  would  have  been  on  interest  1  day,  and  this  is  equiva- 
lent to  having  the  balance,  $75,  on  interest  -97°5°-  of  1  day  =  12  days. 
,  Hence,  the  balance  should  begin  to  draw  interest  12  days  before  May  30; 
that  is,  May  18,  1898. 

2.  Find  the  time  for  the  payment  of  the  balance  of  an 
account  if  the  debit  and  credit  sides,  when  equated,  stand 
as  follows  : 


Dr. 

Due  Feb.  18,  1898,  $950 


Due  Jan.  20, 


Cr. 

1898,  $850 


Solution.     The  difference  between  the  equated  times  is  29  days. 
The  balance  of  account  is  $950  —  $850  =  $100. 

If  the  account  were  settled  at  the  later  date,  Feb.  18,  the  $850 
would  have  been  on  interest  29  days,  which  is  equivalent  to  having  the 
balance,  $100,  on  interest  f  •>-§  of  29  days  =  246  \  days.  Hence,  to 
increase  the  Debit  side  by  an  equal  amount  of  interest,  the  balance 
should  remain  unpaid  247  days  ;  that  is,  the  balance  is  due  Oct.  23, 
1898.     Hence, 


302 


SETTLEMENT    OF    ACCOUNTS. 


527.  To  Find  the  Time  the  Balance  of  an  Account 
Falls  Due, 

Find  the  equated  time  for  each  side  of  the  account. 

Multiply  the  side  of  the  account  that  falls  due  first  by  the 
number  of  days  between  the  dates  of  the  equated  times  of  the 
two  sides,  and  divide  the  product  by  the  balance  of  the  account. 

The  quotient  will  give  the  number  of  days  to  the  maturity 
of  the  balance,  to  be  counted  forward  from  the  later  date  when 
the  smaller  side  falls  due  first,  and  backward  when  the  larger 
side  falls  due  first. 

Note  1.  In  finding  the  equated  time  of  accounts  it  is  customary  to 
neglect  cents  if  less  than  50,  and  if  50  or  more  to  consider  them  as  $1. 
A  fraction  of  a  day  in  the  result  is  rejected  if  less  than  \  ;  if  $  or  more 
it  is  called  a  day. 

Note  2.  When  an  account  is  settled  by  cash  at  any  other  date 
than  that  on  which  the  balance  becomes  due,  the  interest  is  found  on 
the  balance  for  the  interval  between  the  day  of  settlement  and  the  day 
the  balance  is  due,  and  is  added  to  or  deducted  from  the  balance, 
according  as  the  settlement  is  made  after  or  be/ore  the  balance  is  due. 


Exercise  136. 

Find  the  time  for  paying  the  balance  in  the  following 
equated  bills  : 

Average  due.                    Dr.  Average  due.                        Cr. 

1.  May  17,  1897  .  .  .  $950  Apr.  12,  1897  ....  $1000 

2.  Apr.  12,  1897  .  .  .  $950  May  17,  1897  ....  $1000 

3.  May  30,  1898   .  .  $1000  June  23,  1898  ....     $920 

4.  July  6,  1897  .  .  .     $500  Apr.  14,  1897  ....     $480 

5.  Aug.  13,  1897  .  .     $875  Sept.  13,  1897  ....     $600 

6.  May  28,  1898  .  .     $500  June  4,  1898    ....     $550 

7.  Apr.  4,  1898  .  .  .     $400  June  6,  1898    ....     $300 

8.  Mar.  12,  1898  .  .     $750      Feb.  4,  1898 $500 

9.  Feb.  4,  1898  .  .  .     $750  Mar.  12,  1898  ....     $500 


SETTLEMENT  OP  ACCOUNTS. 


303 


528.  The  common  method  of  finding  the  balance  of  an 
account  is  to  compute  the  interest  on  each  item,  from  its 
date  to  the  day  of  settlement,  reckoning  the  time  in  days. 


Dr.       Int. 


Apr.    8.  To  cash,       $250 
May  31.         "  380 

July  20.         "  ■  420 

Oct.  10.  To  bal.  acct.  210 

"10.       "        int. _ 

$1260 


$7.71 

8.36 

5.74 

8.54 

$30.36 

1897. 


Cr. 


Apr.    4.    Bymdse.,$300 
May  19.  "  350 

June  9.  "  610 

Settled  Oct.  10, 1897. 


$1260 


Int. 

$9.45 

8.40 

12.50 


$30.35 


Hence,  the  cash  balance  =  $210  +  $8.54,  or  $218.54. 


Note.     When  the  balance  of  account  and  the  balance  of  interest 
fall  on  opposite  sides,  the  cash  balance  is  their  difference. 

Exercise  137. 

Find  the  cash  balance  of  the  following  accounts,  reckon- 
ing interest  at  6%  : 


Apr.    5. 

"  27. 

June  1. 


To  mdse., 


Dr. 

$250.00 

610.00 

200.00 


1897.  CR. 

Apr.  20.     By  cash,  $200.00 

"   30.  "  500.00 

June  4.  "  400.00 

Settled  June  19,  1897. 


1897.  Dr. 

Jan.  15.  To  mdse.  3  mo.,  $250.00 

Feb.  25.  "          "           98.50 

Mar.    8.  "         "          300.00 

3. 

1897.  Dr. 

Jan.    2.  To  mdse.  60  dy.,  $100.00 


Mar.  8. 
May  10. 
June  2. 


200.00 

30  dy.,   150.00 

95.00 


1897. 

Cr. 

Apr.  26. 

By  cash, 

$150.00 

May  17. 

u 

150.00 

July    7. 

t< 

200.00 

Settled  Oct. 

11,  1897. 

1897. 

Cr. 

Feb.  25. 

By  cash, 

$100.00 

Mar.  22. 

(i 

150.00 

June  21. 

u 

200.00 

Settled  Aug.  2,  1897. 


304 


SAVINGS   BANKS   ACCOUNTS. 


Savings  Banks  Accounts. 

529.  Savings  banks  receive  money  on  deposit,  and  pay 
depositors  compound  interest,  adding  the  interest  to  the 
principal  every  three  months,  six  months,  or  twelve  months. 

530.  The  interval  between  the  dates  at  which  interest 
is  computed  is  called  an  Interest  Term. 

Interest  is  added  at  the  end  of  every  interest  term,  com- 
puted on  the  smallest  balance  on  deposit  at  any  time  during 
the  whole  interest  term. 

Each  depositor  lias  a  bank  book,  in  which  is  recorded 
every  sum  deposited,  every  sum  withdrawn,  and  the  interest 
due  at  the  end  of  each  interest  term. 

531.  Example.  Find  the  balance  on  deposit  Oct.  1,1897, 
on  the  following  account,  interest  4%,  reckoned  quarterly  : 

Deposited  Jan.  1,  1897,  $50  ;  Feb.  4,  1897,  340  ;  May 
6, 1897,  $60 ;  Aug.  3, 1897,  $40.  Withdrawn  Mar.  3, 1897, 
$20  ;  Apr.  22,  1897,  $30  ;  June  19,  1897,  $25  ;  Sept.  20, 
1897,  $40. 

Statement. 


Date. 

Deposited. 

Withdrawn. 

Interest. 

Bajlance. 

1897. 

Jan.  1, 

$50 

00 

$50 

00 

Feb.  4, 

40 

00 

90 

00 

Mar.  3. 

$20 

00 

70 

00 

Apr.  1, 

$0 

50 

70 

50 

Apr.  22, 

30 

00 

40 

50 

May  6, 

60 

00 

100 

50 

June  19, 

25 

00 

75 

50 

July  1, 

0 

40 

75 

90 

Aug.3, 

40 

00 

115 

90 

Sept.  20, 

40 

00 

75 

90 

Oct.  1, 

0 

76 

76 

66 

SAVINGS   BANKS   ACCOUNTS.  305 

The  smallest  sum  on  deposit  during  the  first  interest  term  was  $50. 
The  interest  on  $50  for  3  mo.  at  4%  is  $0.50,  which,  added  to  the 
balance  on  deposit,  makes  $70.50. 

The  smallest  sum  on  deposit  during  the  second  interest  term  was 
$40.50.  The  interest  on  $40.50  for  3  mo.  at  4%  is  $0.40,  which,  added 
to  the  balance  on  deposit,  makes  $75.90. 

The  smallest  sum  on  deposit  during  the  third  interest  term  was 
$75.90.  The  interest  on  $75.90  for  3  mo.  at  4%  is  $0.76,  which,  added 
to  the  balance  on  deposit  Oct.  1,  1897,  makes  $76.66. 


Exercise  138. 

Find  the  balance  on  deposit  Jan.  1,  1898,  on  the  follow- 
ing account : 

1.  Interest  being  4%,  computed  quarterly.     Deposited 
Jan.  1,  1897,  $125  ;   Mar.  22,  1897,  $40  ;   June  8,  1897, ' 
$35 ;   July  30,  1897,  $85  ;    Sept.  24,  1897,  $65.     With- 
drawn Apr.  2,  1897,  $110  ;   June  30,  1897,  $40  ;   Oct.  22, 
1897,  $10  ;   Dec.  17,  1897,  $25. 

2.  Interest  being  3%,  computed  quarterly.  Deposited 
Jan.  1,  1897,  $200  ;  Feb.  14,  1897,  $125  ;  Mar.  10,  1897, 
$75  ;  May  31,  1897,  $50  ;  Aug.  2,  1897,  $100.  With- 
drawn May  7,  1897,  $25  ;  June  22,  1897,  $40  ;  Oct.  2, 
1897,  $50  ;   Nov.  4,  1897,  $65  ;   Dec.  14,  1897,  $75. 

3t  Interest  being  3  °f0,  computed  semi-annually.  De- 
posited Jan.  1,  1897,  $425  ;  May  10, 1897,  $15  ;  Sept.  24, 
1897,  $200;  Oct.  5,  1897,  $25;  Nov.  15,  1897,  $65. 
Withdrawn  Feb.  1,  1897,  $25  ;  Mar.  20,  1897,  $45  ;  Aug. 
2,  1897,  $50  ;   Aug.  28,  1897,  $125  ;   Dec.  10,  1897,  $100. 

4.  Interest  being  3%,  computed  annually.  Deposited 
Jan.  1,  1897,  $266.50  ;  May  3,  1897,  $122.50 ;  Aug.  2, 
1897,  $57  ;  Aug.  9,  1897,  $108  ;  Sept.  4,  1897,  $64.50. 
Withdrawn  June  15,  1897,  $40  ;  Oct.  8,  1897,  $75  ;  Nov. 
1,  1897,  $60  ;   Dec.  4,  1897,  $85  ;   Dec.  20,  1897,  $142. 


CHAPTER   XV. 
POWERS  AND  SOOTS. 

532.  The  square  of  a  number  is  the  product  of  two 
factors,  each  equal  to  the  number  (§  69). 

Thus,  the  squares  of  1,  2,  3,    4,    6,    6,    7,    8,    9,    10, 
are  1,  4,  9,  16,  25,  36,  49,  64,  81,  100. 

533.  The  square  root  of  a  number  is  one  of  the  two 
equal  factors  of  the  number  (§  69). 

Thus,  the  square  roots  of  1,  4,  9,  16,  25,  36,  49,  64,  81,  100, 
are  1,  2,  3,    4,    5,    6,    7,    8,    9,    10. 

534.  The  square  root  of  a  number  is  indicated  by  the 
radical  sign  ^/,  or  by  the  fractional  exponent  £. 

Thus,  V27,  or  27*,  means  the  square  root  of  27. 

535.  Since   35  =  30  +  5,   the   square   of   35   may  be 
obtained  as  follows  : 


30+5 

30+5 

302  =    900 

302  +     (30X5) 

2  (30  X  5)  =   300 

+     (30X5) +  5* 

52  =     25 

302  +  2(30X5)  +  5* 

352  =  1225 

536.    Since  every  number  of  two  or  more  figures  may 
be  regarded  as  composed  of  tens  and  units,  if  we  represent 
the  number  of  tens  by  a  and  the  number  of  units  by  b, 
(a  +  bf  =  a2  +  2ab  +  b*.     Hence, 

The  square  of  a  member  «  equal  to  the  square  of  the  tens, 
plus  twice  the  tens  multiplied  by  the  units,  plus  the  square  of 
the  units. 


POWERS   AND   ROOTS.  307 

537.  The  first  step  in  extracting  the  square  root  of  a 
number  is  to  separate  the  figures  of  the  number  into  groups. 

Since  1  =  l2,  100  =  102,  10,000  =  1002,  and  so  on,  it  is  evident  that 
the  square  root  of  any  number  between  1  and  100  lies  between  1  and 
10  ;  of  any  number  between  100  and  10,000  lies  between  10  and  100. 
In  other  words,  the  square  root  of  any  integral  number  expressed  by 
one  or  two  figures  is  a  number  of  one  figure  ;  expressed  by  three  or 
four  figures  is  a  number  of  two  figures,  and  so  on. 

If,  therefore,  an  integral  number  is  divided  into  groups  of  two 
figures  each,  from  the  right  to  the  left,  the  number  of  figures  in  the 
root  will  be  equal  to  the  number  of  groups  of  figures.  The  last  group 
to  the  left  may  have  one  or  two  figures. 

Example.     Find  the  square  root  of  1225. 

Solution.  The  first  group,  12,  contains  the  square  of  the  tens' 
number  of  the  root. 

The  greatest  square  in  12  is  9,  and  the  square  root  of  9  is  3.    Hence, 

3  is  the  tens'  figure  of  the  root. 
12  25(35  The  square  of  the  tens  is  subtracted,  and  the 

_9 remainder  contains  twice  the  tens  X  the  units  +  the 

65 )  3  25  square  of  the  units.     Twice  the  3  tens  is  6  tens,  and 

3  25  6  tens  is  contained  in  the  32  tens  of  the  remainder 

5  times.  Hence,  5  is  the  units'  figure  of  the  root. 
Since  twice  the  tens  X  the  units  +  the  square  of  the  units  is  equal  to 
(twice  the  tens  +  the  units)  X  the  units,  the  five  units  are  annexed  to 
the  6  tens,  and  the  result,  65,  is  multiplied  by  5. 

538.  The  same  method  will  apply  to  numbers  of  more 
than  two  groups  of  figures,  by  considering  the  part  of  the 
root  already  found  as  so  many  tens  with  respect  to  the  next 
figure  of  the  root. 

Example.     Extract  the  square  root  of  7,890,481. 

7  89  04  81  ( 2809  Solution.     When  the  third  group,  04,  is 

_4 brought  down,  and  the  divisor,  56,  formed,  the 

48 )  3  89  next  figure  of  the  root  is  0,  because  56  is  not 

3  84 contained  in  50.     Therefore,  0  is  placed  both 

5609 )  5  04  81  in  the  root  and  the  divisor,  and  the  next  group, 

5  04  81  81,  is  brought  down. 


308  POWERS   AND   ROOTS. 

539.  If  the  square  root  of  a  number  has  decimal  places, 
the  number  itself  will  have  twice  as  many. 

Thus,  if  0.11  is  the  square  root  of  some  number,  the  number  will 
be  (0.11  )2  =  0.11  X  0.11  =  0.0121.  Hence,  if  a  given  number  contains 
a  decimal,  we  divide  it  into  groups  of  two  figures  each,  beginning 
at  the  decimal  point  and  marking  toward  the  left  for  the  integral 
number,  and  toward  the  right  for  the  decimal.  The  last  group  of  the 
decimal  must  have  two  figures,  a  cipher  being  annexed  if  necessary. 

Example.     Extract  the  square  root  of  52.2729. 

62.27  29(7.23 

49 
142  )fl  27  Solution.     It  will  be  seen  from  the  groups  of 

2  q.  figures  that  the  root  will  have  one  integral  and  two 

1443 )   43  29  decimal  places. 

43  29 

540.  If  a  number  is  not  a  perfect  square,  ciphers  may 
be  annexed,  and  an  approximate  value  of  the  root  found. 

Example.     Extract  the  square  root  of  17  to  six  places. 

17.00  00  00(4.123106 
16 
81 )  1  00  Solution.     In  this  example,  after  finding 

81  four  figures  of  the  root,  the  other  three  are 

822 )   19  00  found  by  common  division. 

16  44  The  rule  in  such  cases  is  that  one  less  than 

8243 )  2  56  00  the  number  of  figures  already  obtained  may  be 

2  47  29  found  without  error  by  division,  the  divisor  to 

8246 )     8  710  be  employed  being  twice  the  part  of  the  root 

8  246  already  found. 

46400 

541.  The  square  root  of  a  common  fraction  is  found  by 
extracting  the  square  root  of  the  numerator  and  of  the 
denominator.  If  the  denominator  is  not  a  perfect  square, 
multiply  both  terms  of  the  fraction  by  a  number  that  will 
make  the  denominator  a  perfect  square,  or  reduce  the  frac- 
tion to  a  decimal  and  extract  the  root  of  the  decimal. 


POWERS    AND   ROOTS.  309 

542.  Rule  for  Square  Root.  Separate  the  number 
into  groups  of  two  figures  each,  beginning  at  the  units. 

Find  the  greatest  square  in  the  left-hand  group  and  write 
its  root  for  the  first  figure  of  the  required  root. 

Square  this  root,  subtract  the  result  from  the  left-hand  group, 
and  to  the  remainder  annex  the  next  group  for  a  dividend. 

For  a  partial  divisor,  double  the  root  already  found,  con- 
sidered as  tens,  and  divide  the  dividend  by  it.  The  quotient 
(or  the  quotient  diminished)  will  be  the  next  figure  of  the 
root. 

To  this  partial  divisor  add  the  last  figure  of  the  root  for 
a  complete  divisor.  Multiply  this  complete  divisor  by  the  last 
figure  of  the  root,  subtract  the  product  from  the  dividend,  and 
to  the  remainder  annex  the  next  group  for  a  new  dividend. 

Proceed  in  this  manner  until  all  the  groups  have  been  thus 
annexed.     The  result  will  be  the  square  root  required. 

Note  1.  If  the  number  is  not  a  perfect  square,  annex  groups  of 
zeros  and  continue  the  process. 

Note  2.  If  the  given  number  contains  a  decimal,  divide  it  into 
groups  of  two  figures  each,  beginning  at  the  decimal  point  and  mark- 
ing toward  the  left  for  the  integral  number  and  toward  the  right  for 
the  decimal  number.  The  last  group  on  the  right  of  the  decimal  must 
contain  two  figures,  a  zero  being  annexed  if  necessary. 


nd 

1  the  square  ] 

Exercise  139. 
root  of  : 

1. 

2916. 

9.    53.7289. 

17. 

8£ 

2. 

7921. 

10.   883.2784. 

18. 

0.9. 

3. 

494,209. 

11.    1.97262025. 

19. 

1- 

4. 

20,164. 

12.    0.0002090916. 

20. 

f- 

5. 

3,345,241. 

13.    2. 

21. 

h 

6. 

125,457.64. 

14.    5. 

22. 

!• 

7. 

47,320,641. 

15.    0.3. 

23. 

3 
4* 

8. 

21,609. 

16.    3i. 

24. 

§• 

310  POWERS   AND   ROOTS. 

Cube  Root. 

543.  The  cube  of  a  number  is  the  product  of  three 
factors,  each  equal  to  the  number  (§  69). 

Thus,  the  cubes  of  1,  2,    3,    4,      5,      6,      7,      8,      9,      10, 
are  1,  8,  27,  64,  125,  216,  343,  512,  729,  1000. 

544.  The  cube  root  of  a  number  is  one  of  the  three 
equal  factors  of  the  number  (§  69). 

Thus,  the  cube  roots  of  1,  8,  27,  64,  125,  216,  343,  512,  729,  1000, 
are  1,  2,    3,    4,      5,      6,      7,      8,      9,      10. 

545.  The  cube  root  of  a  number  is  indicated  by  y/,  a 
small  figure  3  being  written  above  the  radical  sign,  or  by 
the  fractional  exponent  -J. 

Thus,  V343,  or  343*,  means  the  cube  root  of  343. 

546.  Since  35  =  30  +  5,  the  cube  of  35  may  be  obtained 
as  follows : 

30  +5 
30  +5 


302-f-    (30X5)  30s  =  27,000 

+    (30X5)  +  5a  3  (30s  X  5)  =  13,500 

302  +  2(30X5)  +  5a  3(30X5*)=   2,250 
30  +5 53=      125 

303  +  2(302X5)  +    (30  X52)  35s  =  42,875 
-f    (302X5)  +  2(30X52)  +  58 

303  +  3  (302  X  5)  -f  3  (30  X  52)  +  58 

If  we  represent  the  number  of  tens  by  a  and  of  units  by  b 

(a  +  b)s  =  a3  +  3  a2b  +  3  ab2  +  b\     Hence, 

The  cube  of  a  number  is  equal  to  the  cube  of  the  tens,  plus 
three  times  the  product  of  the  square  of  the  tens  by  the  units, 
plus  three  times  the  product  of  the  tens  by  the  square  of  the 
units,  plus  the  cube  of  the  units. 


POWERS    AND    ROOTS.  311 

547.  In  extracting  the  cube  root  of  a  number,  the  first 
step  is  to  separate  the  figures  of  the  number  into  groups. 

Since  1  =  l3,  1000  =  103,  1,000,000  =  1003,  and  so  on,  it  follows 
that  the  cube  root  of  any  integral  number  between  1  and  1000,  that 
is,  of  any  integral  number  that  has  one,  two,  or  three  figures,  is  a 
number  of  one  figure ;  that  the  cube  root  of  any  integral  number 
between  1000  and  1,000,000,  that  is,  of  any  integral -number  that  has 
four,  five,  or  six  figures,  is  a  number  of  two  figures,  and  so  on. 

If,  therefore,  an  integral  number  is  divided  into  groups  of  three 
figures  each,  from  right  to  left,  the  number  of  figures  in  the  root  will 
be  equal  to  the  number  of  groups.  The  last  group  to  the  left  may 
consist  of  one,  two,  or  three  figures. 

Example.     Extract  the  cube  root  of  42,875. 

Solution.  Since  42,875  consists  of  two  groups,  the  cube  root  will 
consist  of  two  figures. 

The  first  group,  42,  contains  the  cube  of  the  tens'  number  of  the 
root. 

The  greatest  cube  in  42  is  27,  and  the  cube  root  of  27  is  3.     Hence, 

3  is  the  tens1  figure  of  the  root. 
42  875  (  35  Tlie  remamder»  15,875,  result- 

27  ing  from  subtracting  the  cube  of 


3  X  302  =  2700 

3  X  (30  X  5)  =    450 

52  =      25. 

3175 


15  875 


the  tens,  will  contain  three  times 

the  product  of  the  square  of  the 

tens  by  the  units  +  three  times 

15  gyg  the  product  of  the  tens  by  the 

square  of  the  units  4-  the  cube 

of  the  units. 

Each   of    these  three    parts    contains    the    units'    number  as  a 

factor. 

Hence,  the  15,875  consists  of  two  factors,  one  of.  which  is  the  units' 
number  of  the  root ;  and  the  other  factor  is  three  times  the  square 
of  the  tens  +•  three  times  the  product  of  the  tens  by  the  units  +  the 
square  of  the  units.  The  largest  part  of  this  second  factor  is  three 
times  the  square  of  the  tens. 

If  the  158  hundreds  of  the  remainder  is  divided  by  the  3  X  302  =  27 
hundreds,  the  quotient  will  be  the  units'  number  of  the  root. 

The  second  factor  can  now  be  completed  by  adding  to  the  2700 
3  X  (30  X  5)  =  450  and  52  =  25. 


312 


POWERS    AND    ROOTS. 


548.  The  same  method  will  apply  to  numbers  of  more 
than  two  groups  of  figures,  by  considering  the  part  of  the 
root  already  found  as  so  many  tens  with  respect  to  the  next 
figure  of  the  root. 

Example.     Extract  the  cube  root  of  57,512,456. 

57  512  456(386 


3  X  302  =  2700 

30  512 

3  X  (30  X  8)  =   720 

82=    64 
3484 

27  872 

2  640  456 

3  X  3802  =  433200 

[  X  (380  X  6)  =   6840 

62=    36 

440076 

2  640  -156 

549.  If  the  cube  root  of  a  number  has  decimal  places, 
the  number  itself  will  have  three  times  as  many. 

Thus,  if  0.11  is  the  cube  root  of  a  number,  the  number  is  0.11  X 
0.11  X  0.11  =  0.001331.  Hence,  if  a  given  number  contains  a  decimal, 
we  divide  the  figures  of  the  number  into  groups  of  three  figures  each, 
by  beginning  at  the  decimal  point  and  marking  toward  the  left  for 
the  integral  number,  and  toward  the  right  for  the  decimal.  We  must 
be  careful  to  have  the  last  group  on  the  right  of  the  decimal  point 
contain  three  figures,  annexing  ciphers  when  necessary. 

Example.     Extract  the  cube  root  of  187.149248. 

187.140  248(5.72 
125 


3  X  502  =  7500 

62  149 

3  X  (50  X  7)  =  1050 

72=  49 

8599 

60  193 

1  956  248 

3  X  5702  =  974700 

3  X  (570  X  2)  =  3420 

22=     4 

97815 

it 

1  956  248 

POWERS   AND   ROOTS. 


313 


It  will  be  seen  from  the  groups  of  figures  that  the  root  will  have 
one  integral  and  two  decimal  places,  and  therefore  the  decimal  point 
must  he  placed  in  the  root  as  soon  as  one  figure  of  the  root  is  obtained. 

550.  If  the  given  number  is  not  a  perfect  cube,  ciphers 
may  be  annexed,  and  a  value  of  the  root  may  be  found  as 
near  to  the  true  value  as  we  please. 

Example.     Extract  the  cube  root  of  5  to  five  places. 


5.000(1.70997 

1 


3  x 


3  x  102  =  300 

(10  X  7)  =  210 

72  =  _49 

559 

259 


J2] 

559  > 
259  J 


4  000 


3  913 


3  X  17002  =  8670000 
3  X  (1700  X  9)  =     45900 

9-2= 81 

8715981 

45981 

3  X  17092  =  8762043 


} 


87  000  000 


78  443  829 


8  556  1710 

7  885  8387 


670  33230 
613  34301 


After  the  first  two  figures  of  the  root  are  found,  the  next  trial  divi- 
sor is  obtained  by  bringing  down  the  sum  of  the  210  and  49  obtained 
in  completing  the  preceding  divisor,  then  adding  the  three  numbers 
connected  by  the  brace,  and  annexing  two  ciphers  to  the  result. 

It  is  seen  at  a  glance  that,  when  the  trial  divisor  is  increased  by 
3  times  the  17  tens  of  the  root,  it  will  be  greater  than  87,000 ;  so  that 
0  is  placed  in  the  root,  and  3  X  17002  is  obtained  by  annexing  two 
ciphers  to  the  86,700.  Again  :  the  last  divisor  is  obtained  by  bringing 
down  the  sum  of  the  45,900  and  81,  which  were  obtained  in  completing 
the  preceding  divisor,  then  adding  the  three  numbers  connected  by 
the  brace. 

The  last  two  figures  of  the  root  are  found  by  division. 

The  rule  in  such  cases  is  that  two  less  than  the  number  of  figures 
already  obtained  may  be  found  without  error  by  division,  the  divisor  to 
be  employed  being  three  times  the  square  of  the  part  of  the  root  already 
found. 


314  POWERS    AND    ROOTS. 

551.  The  cube  root  of  a  common  fraction  is  found  by- 
taking  the  cube  root  of  the  numerator  and  of  the  denomi- 
nator. If  the  denominator  is  not  a  perfect  cube,  multiply 
both  terms  of  the  fraction  by  a  number  that  will  make  the 
denominator  a  perfect  cube,  or  reduce  the  fraction  to  a  deci- 
mal, and  then  extract  the  cube  root  of  the  decimal. 

552.  Rule  for  Cube  Root.  Separate  the  number  into 
groups  of  three  figures  each,  beginning  at  the  units. 

Find  the  greatest  cube  in  the  left-hand  group  and  write  its 
root  for  the  first  figure  of  the  required  root. 

Cube  this  root,  subtract  the  result  from  the  left-hand 
group,  and  to  the  remainder  annex  the  next  group  for  a 
dividend. 

For  a  partial  divisor,  take  three  times  the  square  of  the 
root  already  found,  considered  as  tens,  and  divide  the  divi- 
dend by  it.  The  quotient  (or  the  quotient  diminished)  will  be 
the  second  figure  of  the  root. 

To  this  partial  divisor  add  three  times  the  product  of  the 
first  figure  of  the  root  considered  as  tens  by  the  second  figure, 
and  also  the  square  of  the  second  figure.  This  sum  will  be 
the  complete  divisor. 

Multiply  the  complete  divisor  by  the  second  figure  of  the 
root,  subtract  the  product  from  the  dividend,  and  to  the 
remainder  annex  the  next  group  for  a  new  dividend. 

Proceed  in  this  manner  until  all  the  groups  have  been 
annexed.     The  result  will  be  the  cube  root  required. 

Note  1.  If  the  number  is  not  a  perfect  cube,  annex  groups  of  zeros 
and  continue  the  process. 

Note  2.  If  the  given  number  contains  a  decimal,  divide  it  into 
groups  of  three  figures  each,  beginning  at  the  decimal  point  and  mark- 
ing toward  the  left  for  the  integral  number  and  toward  the  right  for 
the  decimal  number.  The  last  group  on  the  right  of  the  decimal 
must  contain  three  figures,  zeros  being  annexed  if  necessary. 


POWERS   AND   ROOTS. 


315 


Exercise  140. 
Find  the  cube  root  of  : 

304,957.115891. 

0.007821346625. 

104.600290750613. 

17,183,498,535,125. 

122,615.327232. 

116,400. 

22,406,807. 


1. 

1331. 

8. 

2. 

1728. 

9. 

3. 

12.167. 

10. 

4. 

300.763. 

11. 

5. 

148,877. 

12. 

6. 

2,048,383. 

13. 

7. 

59.776471. 

14. 

15. 

10 

16. 

3* 

17. 

H 

18. 

5. 

19. 

f- 

20. 

n 

21. 

3 
4* 

Geometrical  Representation  of  Square  and  Cube  Roots. 
553.   We  will  illustrate  square  root  by  giving  a  Geo- 


metrical representation  of  the  square  root  of  1225. 


(§  537) 


The  square  root  of  1225  is  35. 
The  square  of  (30  +  5)  =  302  +  2  (30  X  5)  +  52.  (§  536) 

The  302  may  be  represented  by  a  square  (Fig.  1)  30  in.  on  a  side. 
The  2  (30  X  5)  may  be  represented  by  two  strips  30  in.  long  and 
5  in.  wide  of  Fig.  2,  which  are  added  to  two  adjacent  sides  of  Fig.  1. 


Fig.  l. 


Fig.  2. 


Fig.  3. 


The  52  may  be  represented  by  the  small  square  of  Fig.  3  required 
to  make  Fig.  2  a  complete  square. 

In  extracting  the  square  root  of  1225,  the  large  square,  which  is 
30  in.  on  a  side,  is  first  removed,  and  a  surface  of  325  sq.  in.  remains. 

This  surface  consists  of  two  equal  rectangles,  each  30  in.  long,  and 
a  small  square  whose  side  is  equal  to  the  width  of  the  rectangles. 

The  width  of  the  rectangles  is  found  by  dividing  the  325  sq.  in.  by 
the  sum  of  their  lengths ;   that  is,  by  60  in.,  which  gives  5  in. 

Hence,  the  entire  length  of  the  surfaces  added  is  30  in.  +  30  in. 
+  6  in.  =  65  in.,  and  the  width  is  5  in. 

Therefore,  the  total  area  is  (65  X  5)  sq.  in.  =  325  sq.  in. 


316 


POWERS    AND    ROOTS. 


554.   We  will  illustrate  cube  root  by  giving  a  Geomet- 
rical representation  of  the  cube  root  of  42,875. 

The  cube  root  of  42,875  is  35.  (§  547) 

The  cube  of  (30  +  5)  =  30»  +  3  (302  x  5)  +  3  (30  X  52)  +  58.  (§  546) 
The  308  may  be  represented  by  a  cube  whose  edge  is  30  in.  (Fig.  1). 
The  3  (302  X  5)  may  be  represented  by  three  rectangular  solids,  each 

30  in.  long,  30  in.  wide,  and  6  in.  thick,  to  be  added  to  three  adjacent 

faces  of  Fig.  1. 

The  3(30  X  52)  may  be  represented  by  three  equal  rectangular 

solids,  30  in.  long,  5  in.  wide,  and  5  in.  thick,  to  be  added  to  Fig.  2. 
The  58  may  be  represented  by  the  small  cube  required  to  complete 

the  cube  of  Fig.  3. 


S^&\t 


^L 


m 


Fio.  1. 


Fig.  2. 


Fio.  3. 


M 


Fio.  4. 


In  extracting  the  cube  root  of  42,875,  the  large  cube  (Fig.  1),  whose 
edge  is  30  in.,  is  first  removed. 

There  remain  (42,875  —  27,000)  cu.  in.  =  15,875  cu.  in. 

The  greatest  part  of  this  is  contained  in  the  three  rectangular  solids 
which  are  added  to  Fig.  1,  and  are  each  30  in.  long  and  30  in.  wide. 

The  thickness  of  these  solids  is  found  by  dividing  the  15,875  cu.  in. 
by  the  sum  of  the  three  faces,  each  of  which  is  30  in.  square  ;  that  is, 
by  2700  sq.  in.     The  result  is  5  in. 

There  are  also  the  three  rectangular  solids  which  are  added  to 
Fig.  2,  and  which  are  30  in.  long  and  5  in.  wide ;  and  a  cube  which  is 
added  to  Fig.  3,  and  which  is  5  in.  long  and  5  in.  wide. 

Hence,  the  sum  of  the  products  of  two  dimensions  of  all  these 
solids  is 

For  the  larger  rectangular  solids,  3  (30  X  30)  sq.  in.  =  2700  sq.  in. 

For  the  smaller  rectangular  solids,  3  (30  X  5)  sq.  in.  =    450  sq.  in. 

For  the  small  cube,  (5  x  5)  sq.  in.  = 25  sq.  in. 

3175  sq.  in. 

This  number  multiplied  by  the  third  dimension  gives  (5  X  3175) 
cu.  in.  =  15,875  cu.  in. 


CHAPTER   XVI. 

MENSUBATION. 

555.  We  have  already  considered  Areas  of  Rectangles 
and  Circles  ;  and  Volumes  and  Surfaces  of  Rectangular 
Solids,  Spheres,  and  Right  Cylinders. 

556.  A  polygon  is  a  plane  figure  bounded  by  straight  lines. 
A  polygon  of  three  sides  is  a  triangle ;   of  four  sides,  a 

quadrilateral ;  of  five  sides,  a  pentagon ;  of  six  sides,  a 
hexagon;  of  eight  sides,  an  octagon;  of  ten  sides,  a  deca- 
gon;   of  twelve  sides,  a  dodecagon  ;   and  so  on. 

557.  The  area  of  any  polygon  may  be  found  by  divid- 
ing it  into  triangles  and  finding  the  sum  of  their  areas. 

558.  A  vertex  of  a  polygon  is  the  point  of  intersection 
of  two  adjacent  sides. 

559.  A  diagonal  of  a  polygon  is  a  straight  line  joining 
any  two  vertices  not  adjacent. 


Polygon.  Regular  Polygon. 

560.  A  regular  polygon  is  a  polygon  with  all  its  sides 
equal  and  all  its  angles  equal.  The  centre  of  a  regular 
polygon  is  a  point  equidistant  from  the  vertices  and  also 
equidistant  from  the  sides.  The  radius  of  a  regular  poly- 
gon is  the  distance  from  the  centre  to  any  vertex. 


318  MENSURATION. 

The  radii  of  a  regular  polygon  divide  the  polygon  into 
equal  isosceles  triangles  ;  that  is,  into  triangles  having  two 
sides  equal.  The  apothem  of  a  regular  polygon  is  the  dis- 
tance from  the  centre  to  any  side. 

561.  The  area  of  a  regular  polygon  =  £  (perimeter  X 
apothem). 

562.  The  apothem  of  a  regular  polygon  bears  a  constant 
ratio  to  one  side. 

The  following  table  shows  the  ratio  of  the  apothem  to 
one  side  in  the  most  common  regular  polygons  : 

Triangle 0.2887:1.  Heptagon  ....  1.0382:1. 

Quadrilateral  .  .  0.5000  : 1.  Octagon 1.2071 : 1. 

Pentagon  ....  0.6882  : 1.  Decagon    ....  1.5388  : 1. 

Hexagon    ....  0.8660:1.  Dodecagon  .  .  .  1.8660:1. 

Quadrilaterals. 


LA 


Trapezium.  Trapezoid.  Parallelogram. 

563.  A  trapezium  is  a  quadrilateral  with  no  two  of  its 
sides  parallel. 

Note.     Two  lines  are  parallel  if  all  points  of  one  are  equally  dis- 
tant from  the  other. 

564.  A  trapezoid  is  a  quadrilateral  with  two  of  its  sides 
parallel,  but  the  other  two  sides  not  parallel. 

565.  A  parallelogram  is  a  quadrilateral  with  its  oppo- 
site sides  parallel. 

566.  A  rhomboid  is  a  parallelogram  with  its  angles  not 
right  angles. 

567.  A  rhombus  is  a  parallelogram  with  its  angles  not 
right  angles,  but  with  all  its  sides  equal. 


MENSURATION. 


319 


Parallelograms. 


Rhomboid. 


Rhombus. 


Square. 


Rectangle. 


568.  The  altitude  of  a  parallelogram  or  of  a  trapezoid 
is  the  shortest  distance  between  its  parallel  sides  regarded 
as  bases. 

569.  The  area  of  any  parallelogram  =  base  X  altitude. 

570.  The  area  of  a  rhombus  also  =  half  the  product  of 
its  diagonals. 

571.  The  area  of  a  trapezoid  =  £  (sum  of  bases  X  alti- 
tude). 

Triangles. 


Right. 


Isosceles. 


Equilateral. 


Scalene. 


572.  A  right  triangle  is  a  triangle  one  of  whose  angles 
is  a  right  angle.  The  hypotenuse  is  the  side  opposite  the 
right  angle,  and  the  other  two  sides,  called  legs,  are  the 
base  and  the  perpendicular. 

573.  Other  kinds  of  triangles  are,  isosceles,  with  two 
sides  equal ;  equilateral,  with  three  sides  equal ;  scalene, 
with  no  two  sides  equal.  The  altitude  of  a  triangle  is  the 
shortest  distance  from  the  vertex  to  the  base  or  the  base 
produced. 

574.  When  the  base  and  altitude  are  given, 

The  area  of  the  triangle  =  \  (base  X  altitude). 


320 


MENSURATION. 


575.  When  the  sides  of  a  triangle  are  given, 

The  area  of  the  triangle  is  the  square  root  of  the  product 
of  half  the  sum  of  the  sides  multiplied  in  succession  by  the 
three  remainders  obtained  by  subtracting  each  side  separately 
from  the  half  sum  of  the  sides. 

576.  In  a  right  triangle,  the  square  of  the  hypote- 
nuse is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides. 

The  hypotenuse  is,  therefore, 
equal  to  the  square  root  of  the 
sum  of  the  squares  of  the  other 
two  sides  ;  and  either  leg  is  equal 
to  the  square  root  of  the  differ- 
ence of  the  squares  of  the  hypote- 
nuse and  the  other  leg. 

577.  Examples.     Find  the  area  of : 

1 .  A  regular  hexagon,  each  side  of  which  is  3  in. 

Solution.     The  apothem  =  0.8660  X  3  in.  =  2.598  in.; 
and  the  perimeter  =  6  X  3  in.  =  18  in. 

Therefore,  the  area  =  i(18  X  2.598)  sq.  in.  =  23.382  sq.  in. 

2.  A  parallelogram,  base  12  in.,  altitude  7  in. 
Solution.     The  area  =  (12  x  7)  sq.  in.  =  84  sq.  in. 

3.  A  trapezoid,  if  its  altitude  is  10  in.,  and  its  parallel 
sides  are  16  in.  and  12  in.,  respectively. 

Solution.     The  sum  of  the  bases  is  16  in.  +  12  in.  =  28  in. 
Therefore,  the  area  =  \  (28  X  10)  sq.  in.  =  140  sq.  in. 

4.  A  triangle,  base  12  in.,  altitude  8  in. 
Solution.    The  area  =  i  (12  x  8)  sq.  in.  =  48  sq.  in. 

5.  A  triangle,  sides  5  in.,  6  in.,  7  in. 

Solution.     The  half  sum  of  the  sides  is  I  (6  +  6  +  7)  in.,  or  9  in. 
Hence,  the  area  =  V9X4X8X2  sq.  in.  =  V216  sq.  in.  =  14.696  sq.  in. 


MENSURATION.  321 

6.  The  base  of  a  right  triangle  is  20  ft.,  and  the  perpen- 
dicular is  15  ft.     Find  the  hypotenuse. 

Solution.      V202  +  152  =  V400  +  225  =  V625  =  25. 
Therefore,  the  length  of  the  hypotenuse  is  25  ft. 

7.  The  base  of  a  right  triangle  is  16  ft.,  and  the  hypote- 
nuse is  20  ft.     Find  the  perpendicular. 

Solution.     V202  —  162  =  V400  -  256  =  Vl44  =  12. 
Therefore,  the  length  of  the  perpendicular  is  12  ft. 

Exercise  141. 
Find  the  area  of  : 

1.  A  parallelogram,  base  18  in.,  altitude  11  in. 

2.  A  triangle,  base  16  in.,  altitude  12  in. 

3.  A  rectangle,  base  24  in.,  altitude  18  in. 

4.  A  square,  side  18  in. 

5.  A  rhombus,  diagonals  8  in.  and  10  in. 

6.  A  triangle,  sides  12  in.,  11  in.,  and  10  in.,  respectively. 

7.  A  regular  hexagon,  side  4  in. 

8.  A  regular  octagon,  side  2  in. 

9.  A  triangle,  base  185  yd.,  altitude  154  yd. 

10.  A  square,  side  212  yd. 

11.  A  rectangle,  base  106  yd.,  altitude  66  yd. 
12,.   A  parallelogram,  base  24  ft.,  altitude  18  ft. 

13.  An  equilateral  triangle,  side  132  yd. 

14.  A  right  triangle,  base  164  ft.,  perpendicular  150  ft. 

15.  A  regular  pentagon,  side  5 J  in. 

16.  A  parallelogram,  base  122  yd.,  altitude  76  yd. 

17.  A  regular  decagon,  side  2\  in. 

18.  A  triangle,  base  82cm,  altitude  51cm. 

19.  A  rhombus,  diagonals  16  ft.  and  12  ft. 

20.  A  circle,  diameter  72  ft. 

21.  A  trapezoid,  parallel  sides  106  ft.  and  56  ft.,  respec- 
tively, altitude  48  ft. 


322  MENSURATION. 

22.  Find  the  number  of  hektars  in  a  triangular  field, 
one  side  of  which  is  82.1m,  and  the  distance  to  this  side 
from  the  opposite  corner  47.3m. 

23.  Find  the  number  of  acres  in  a  triangular  field,  one 
side  of  which  is  343.6  ft.,  and  the  distance  to  this  side 
from  the  opposite  cornel-  163.2  ft. 

24.  Find  the  area  of  a  circle  that  has  a  radius  of  10  in. ; 
of  a  circle  that  has  a  diameter  of  10  ft.;  of  a  circle  that 
has  a  circumference  of  30  in. 

25.  A  horse  is  tied  by  a  rope  27. 8m  long;  over  what 
part  of  a  hektar  can  he  graze  ? 

26.  How  many  square  feet  in  a  circle  that  has  a  diam- 
eter of  17#  yd.? 

27.  How  many  square  feet  in  a  circle  that  has  a  circum- 
ference of  117  yd.? 

28.  Find  the  area  of  a  triangle  whose  sides  are  73  ft., 
57  ft.,  and  48  ft. 

29.  Find  the  number  of  hektars  in  a  triangular  field 
whose  sides  are  37.5m,  91. 7m,  and  78.9m. 

30.  Find  the  number  of  hektars  in  a  triangular  field 
whose  sides  are  67.5m,  81.2m,  and  102.7m. 

31.  Find  the  number  of  acres  in  a  triangular  field  whose 
sides  are  227  ft.,  342  ft.,  and  416  ft. 

32.  Find  the  number  of  acres  in  a  triangular  field  whose 
sides  are  79.08  ch.,  57.03  ch.,  and  102.19  ch. 

33.  Find  the  number  of  square  rods  in  a  triangle  whose 
sides  are  7  rd.  2  yd. ;  6  rd.  5  yd. ;  and  9  rd.  4£  ft. 

34.  One  diagonal  of  a  trapezium  is  10  rd.,  and  the  per- 
pendiculars upon  it  from  the  opposite  corners  are  6  rd.  and 
8  rd.     Find  the  area. 

35.  Find  the  area  of  a  lot  of  land  in  the  shape  of  a 
trapezium,  if  one  diagonal  is  108  ft.,  and  the  perpendicu- 
lars upon  it  from  the  opposite  corners  are  55  ft.  and 
60  ft. 


MENSURATION.  323 

36.  What  is  the  area  of  the  ground  covered  by  a  tent, 
the  base  of  which  is  a  regular  heptagon  25  ft.  on  a  side  ? 

37.  How  many  paving  stones  will  be  required  to  pave  a 
rectangular  court  60  ft.  long  and  40  ft.  wide,  if  each  stone 
is  in  the  shape  of  a  regular  hexagon  5  in.  on  a  side  ? 

38.  At  $225  an  acre,  what  is  the  value  of  a  field  in  the 
shape  of  a  regular  pentagon  250  yd.  on  a  side  ? 

39.  A  rectangular  field  100  yd.  wide  contains  3£  A. 
What  is  its  length  ? 

40.  The  dimensions  of  a  rectangle  are  45  yd.  and  28  yd. 
What  is  the  length  of  its  diagonal  ? 

41.  A  field  has  the  shape  of  a  right  triangle,  and  the  two 
legs  are  75  yd.  and  60  yd.,  respectively.  What  decimal  of 
an  acre  does  the  field  contain  ? 

42.  Compare  the  areas  of  a  square  and  an  equilateral 
triangle,  if  the  perimeter  of  each  is  60  ft. 

43.  Find  the  area  of  a  field  in  the  shape  of  a  trapezoid, 
if  the  altitude  is  240  yd.,  and  the  parallel  sides  are  510  yd. 
and  725  yd.,  respectively. 

44.  The  legs  of  a  right  triangle  are  each  equal  to  12  ft. 
Find  the  hypotenuse. 

45.  A  city  lot  in  the  shape  of  a  right  triangle  has  for  its 
base  119  ft.,  and  for  its  perpendicular  120  ft.  Find  the 
area  and  the  hypotenuse  of  the  lot. 

46.  Find  the  base  and  the  area  of  a  right  triangle,  hypot- 
enuse 130  yd.,  and  perpendicular  112  yd. 

47.  Find  the  perpendicular  and  the  area  of  a  right  tri- 
angle, hypotenuse  164  ft.,  and  base  160  ft. 

48.  Find  the  hypotenuse  and  the  area  of  a  right  triangle, 
base  100  yd.,  and  perpendicular  105  yd. 

49.  Find  the  hypotenuse  and  the  area  of  a  right  triangle, 
base  96  ft.,  and  perpendicular  110  ft. 

50.  Find  the  area  of  a  field  in  the  shape  of  a  right  tri- 
angle, if  the  hypotenuse  is  709  yd.,  and  one  leg  660  yd. 


324  MENSURATION. 

51.  A  rectangular  field  is  345  yd.  long  and  152  yd.  wide. 
What  is  the  length  of  its  diagonal  ? 

52.  The  legs  of  a  right  triangle  are  44  ft.  4  in.  and  13  ft. 
9  in.,  respectively.     Find  the  length  of  its  hypotenuse. 

53.  The  hypotenuse  of  a  right  triangle  is  7  ft.  1  in.,  and 
one  leg  is  6  ft.  5  in.     Find  the  other  leg  and  the  area. 

54.  The  hypotenuse  of  a  right  triangle  is  3  ft.  1  in.,  and 
one  leg  is  2  ft.  11  in.     Find  the  other  leg  and  the  area. 

55.  The  area  of  a  lot  in  the  shape  of  a  right  triangle  is 
1560  sq.  yd.,  and  the  base  is  80  yd.  Find  the  perpendicu- 
lar and  the  hypotenuse. 

56.  The  area  of  a  right  triangle  is  60  sq.  in.,  and  one  leg 
is  8  in.     Find  the  hypotenuse  and  the  other  leg. 

57.  The  length  and  diagonal  of  a  rectangular  field  are 
60  rd.  and  65  rd.,  respectively.     What  is  its  area? 

58.  What  is  the  length  of  a  side  of  a  square  that  con- 
tains 390,625  sq.  ft.? 

59.  Express  to  six  places  of  decimals  the  length  of  the 
diagonal  of  a  square  in  terms  of  a  side. 

60.  The  hypotenuse  of  a  right  triangle  is  95  ft.,  and  the 
two  legs  are  as  3  to  4.     Find  the  legs  and  the  area. 

61.  St.  Mark's  Square  in  Venice  has  the  shape  of  a  trape- 
zoid. The  parallel  sides  are  61  yd.  and  90  yd.,  respec- 
tively, and  the  altitude  is  192  yd.     What  is  its  area  ? 

62.  The  perimeter  of  a  regular  hexagon  is  45  in.  Find 
its  area. 

63.  A  circular  pond  contains  12  acres.  Express  its 
diameter  in  feet. 

Note.  Multiply  the  area  in  square  feet  by  0.31831,  and  take  the 
square  root  for  the  radius. 

64.  What  is  the  diameter  of  a  circle  whose  area  is 
1262  sq.  ft.? 

65.  What  is  the  diameter  of  a  circle  whose  area  is 
2206  sq.  ft.? 


MENSURATION. 


325 


Solids. 

578.  A  right  prism  is  a  solid  bounded  by  two  equal 
parallel  polygons,  called  the  bases,  and  by  rectangles,  called 
the  lateral  faces.  The  altitude  of  a  right  prism  is  the 
shortest  distance  between  its  bases. 


Right  Prism. 


Regular  Pyramid. 


Right  Cone. 


579.  A  regular  pyramid  is  a  solid  bounded  by  a  regular 
polygon,  called  the  base,  and  by  isosceles  triangles,  called 
the  lateral  faces.  These  triangles  all  terminate  in  a  point 
called  the  vertex  of  the  pyramid.  The  altitude  of  a  regular 
pyramid  is  the  shortest  distance  from  its  vertex  to  its  base. 
The  slant  height  is  the  altitude  of  the  lateral  faces. 

580.  A  cone  is  a  solid  bounded  by  a  circle,  called  the 
base,  and  by  a  curved  surface,  called  the  lateral  surface, 
which  terminates  in  a  point  called  the  vertex. 

581.  A  right  cone  is  a  cone  whose  vertex  is  in  the  per- 
pendicular erected  at  the  centre  of  the  base.  The  altitude 
of  a  right  cone  is  the  shortest  distance  from  its  vertex  to  its 
base.  The  slant  height  of  a  right  cone  is  the  distance  from 
its  vertex  to  the  circumference  of  its  base. 

582.  A  frustum  of  a  regular  pyramid  or  a  frustum 
of  a  right  cone  is  the  part  of  the  pyramid  or  of  the  cone 
left  after  its  top  has  been  cut  off  by  a  plane  parallel  to  its 
base.  The  lateral  faces  of  a  frustum  of  a  regular  pyramid 
are  trapezoids. 


326  MENSURATION. 

583.  The  bases  of  a  frustum  of  a  regular  pyramid  or  of 
a  frustum  of  a  right  cone  are  the  base  of  the  pyramid  or 
cone  and  the  section  made  by  the  cutting  plane.  The 
altitude  of  the  frustum  of  a  regular  pyramid  or  of  the 
frustum  of  a  right  cone  is  the  shortest  distance  between 
its  bases. 


Frustum  of  a  Regular  Pyramid.  Frustum  of  a  Right  Cone. 

584.  The  area  of  the  lateral  surface  of  a  right  prism, 
of  a  regular  pyramid,  or  of  the  frustum  of  a  regular  pyra- 
mid, is  the  sum  of  the  areas  of  its  lateral  faces. 

585.  The  lateral  surface  of  a  right  prism  =  perimeter  of 
base  X  altitude. 

586.  The  volume  of  a  right  prism  =  base  X  altitude. 

587.  The  lateral  surface  of  a  regular  pyramid  =  £  ( per- 
imeter of  base  X  slant  height). 

588.  The  volume  of  a  regular  pyramid  =  £  (base  X  alti- 
tude). 

589.  The  lateral  surface  of  a  right  cone  =  £  (circumfer- 
ence of  base  X  slant  height). 

590.  The  volume  of  a  right  cone  =  J-  (base  X  altitude). 

591.  The  lateral  surface  of  a  frustum  of  a  regular  pyra- 
mid or  of  a  frustum  of  a  right  cone  =  £  (sum  of  perimeters 
of  bases  X  slant  height). 


MENSURATION.  327 

592.  To  find  the  volume  of  a  frustum  of  a  regular  pyra- 
mid or  of  a  frustum  of  a  right  cone,  we  take  the  sum  of  the 
areas  of  its  bases  and  the  square  root  of  their  product ;  and 
multiply  this  sum  by  one  third  the  altitude. 

593.  Examples.  1.  Find  the  lateral  surface  and  the 
volume  of  a  right  prism,  base  a  square  3  ft.  on  a  side,  and 
altitude  5  ft. 

Solution.     The  perimeter  of  base  =  4  X  8  ft.  =  12  ft. 
Hence,  the  lateral  surface  =  (12  X  5)  sq.  ft.  =  60  sq.  ft. 
The  area  of  base  =  (3x3)  sq.  ft.  —  9  sq.  ft. 
Hence,  the  volume  =  (9X5)  cu.  ft.  =  45  cu.  ft. 

2.  Find  the  lateral  surface  and  the  volume  of  a  regular 
pyramid,  base  a  square  6  ft.  on  a  side,  and  altitude  4  ft. 

Solution.     The  perimeter  of  base  =  4x6  ft.  =  24_ft. 
The  slant  height  =  V3a  +  42  ft.  =  V9  +  16  ft  =  V25  ft.  =  5  ft. 
Hence,  the  lateral  surface  =  i(24  X  5)  sq.  ft.  =  60  sq.  ft. 
The  area  of  base  =  (6x6)  sq.  ft.  =  36  sq.  ft. 
Hence,  the  volume  =  £  (36  X  4)  cu.  ft.  =  48  cu.  ft. 

3.  Find  the  volume  of  a  frustum  of  a  regular  pyramid, 
bases  squares,  5  ft.  and  3  ft.,  respectively,  on  a  side,  and 
altitude  6  ft. 

Solution.     The  areas  of  the  bases  are  25  sq.  ft.  and  9  sq.  ft., 
respectively  ;   and  the  square  root  of  their  product  is  15  sq.  ft. 
Hence,  the  volume  =  £  X  6  (25  +  9  +  15)  cu.  ft.  =  98  cu.  ft. 

Exercise  142. 

1.  Find  the  volume  of  a  triangular  prism,  height  11  in., 
and  sides  of  the  ends  2  in.,  3  in.,  and  4  in.,  respectively. 

2.  Find  the  capacity  in  bushels  of  a  bin  6  ft.  long,  the 
end  of  which  is  a  square  3  ft.  3  in.  on  a  side. 

3.  Find  the  lateral  surface  and  the  volume  of  a  regular 
pyramid,  base  a  regular  hexagon  9  in.  on  a  side,  altitude 
40  in.,  and  slant  height  40.75  in. 


328  MENSURATION. 

4.  Find  the  number  of  cubic  yards  in  a  prism,  base  a 
square  200  ft.  on  a  side,  height  40  ft. 

5.  How  many  square  yards  of  canvas  are  required  for 
a  conical  tent  9  ft.  11  in.  high,  diameter  of  base  20  ft.? 

6.  Find  the  volume  and  the  lateral  surface  of  a  frustum 
of  a  regular  pyramid,  bases  squares,  24  in.  and  12  in.  on  a 
side,  respectively,  altitude  17£  in.,  slant  height  18£  in. 

7.  Find  the  volume  and  the  lateral  surface  of  a  frustum 
of  a  right  cone,  radii  of  bases  50cm  and  30cm,  respectively, 
altitude  48cm,  and  slant  height  52cm. 

8.  Find  the  volume  and  the  surface  of  a  sphere  whose 
diameter  is  17.2cm. 

9.  A  right  cylinder  is  3  ft.  2  in.  in  diameter  and  4  ft. 
6  in.  high.     Find  its  volume  and  its  lateral  surface. 

10.  Find  the  length  of  an  edge  of  a  cubical  vessel  that 
will  hold  a  ton  of  water. 

11.  A  rectangular  tank  6  ft.  long  and  4£  ft.  wide  holds 
108  cu.  ft.  of  water.     What  is  the  height  of  the  tank  ? 

12.  Find  the  total  surface  of  a  regular  pyramid,  base  a 
square  5  ft.  on  a  side,  and  slant  height  20  ft. 

1 3.  The  circumference  of  the  base  of  a  right  cone  is  12  ft., 
and  the  height  of  the  cone  is  12  ft.     Find  the  volume. 

14.  Find  the  surface  of  a  megaphone  in  the  shape  of  a 
frustum  of  a  right  cone,  diameters  of  the  upper  and  lower 
bases  24  in.  and  3  in.,  respectively,  slant  height  30  in. 

15.  Find  the  difference  between  the  volume  of  a  frus- 
tum of  a  regular  pyramid,  bases  squares,  8  ft.  and  6  ft., 
respectively,  on  a  side  and  altitude  9  ft.,  and  the  volume 
of  a  right  prism,  base  a  square  7  ft.  on  a  side,  altitude  9  ft. 

16.  Find  the  surface  and  the  volume  of  a  sphere  whose 
diameter  is  28  in. 

17.  Find  the  ratio  of  the  volume  of  a  cube  of  wood  15  in. 
on  an  edge  to  the  volume  of  the  largest  sphere  that  can  be 
turned  from  it.     Find  the  ratio  of  their  surfaces. 


MENSURATION.  329 

18.  Find  the  ratio  of  the  volume  of  a  cube  of  wood  to 
the  volume  of  the  largest  right  cylinder  that  can  be  turned 
from  it.     Find  the  ratio  of  their  surfaces. 

19.  Find  the  ratio  of  the  volume  of  a  right  cylinder  of 
wood  to  the  volume  of  the  largest  right  cone  that  can  be 
turned  from  it.     Find  the  ratio  of  their  lateral  surfaces. 

20.  Find  the  length  of  an  edge  of  a  cube  that  contains 
100  cu.  in. 

21.  The  Great  Pyramid  of  Egypt  was  originally  made 
in  the  form  of  a  regular  pyramid,  altitude  480£  ft.,  and 
base  a  square  764  ft.  on  a  side.  Find  in  acres  the  area  of 
the  ground  covered  by  the  pyramid.  Find  in  cubic  yards 
the  volume,  and  in  square  yards  the  lateral  surface  of  the 
pyramid. 

22.  The  mast  of  a  ship  is  80  ft.  high,  and  the  diameters 
of  its  ends  are  4  ft.  6  in.  and  2  ft.,  respectively.  Find  its 
value  at  75  cents  a  cubic  foot. 

23.  A  spherical  shot  6  in.  in  diameter  is  melted  and  cast 
into  a  cylinder  3  in.  in  diameter.  What  is  the  height  of 
this  cylinder  ? 

24.  A  cylindrical  pail  14  in.  High,  holds  2  cu.  ft.  of 
water.     What  is  the  diameter  of  its  base  ? 

25.  A  regular  pyramid  14  in.  high  has  for  its  base  an 
equilateral  triangle  6  in.  on  a  side.     What  is  its  volume  ? 

26.  A  right  prism  8  in.  high  has  for  its  base  a  trapezoid 
whose  altitude  is  4  in.,  and  whose  parallel  sides  are  5  in. 
and  3  in.,  respectively.  What  is  the  volume  and  the  total 
surface  of  the  prism  ? 

27.  A  rectangular  room  is  18  ft.  long,  16  ft.  wide,  and 
12  ft.  high.  What  is  the  distance  from  the  upper  right- 
hand  corner  to  the  opposite  lower  left-hand  corner  ? 

28.  A  conical  spire  40  ft.  high  has  a  base  15  ft.  in  diam- 
eter. Find  the  cost  at  5  cents  a  square  inch  of  gilding  the 
spire. 


330  MENSURATION. 

594.  Similar  Figures.  Figures  that  have  the  same 
shape  are  called  similar  figures. 

595.  The  corresponding  lines  of  similar  figures  are  pro- 
portional. 

596.  The  surfaces  of  similar  figures  are  to  each  other  as 
the  squares  of  their  corresponding  dimensions;  and  their 
volumes  are  to  each  other  as  the  cubes  of  their  cori'esponding 
dimensions. 

597.  The  corresponding  dimensions  of  similar  figures  are 
to  each  other  as  the  square  roots  of  their  surfaces,  or  as  the 
cube  roots  of  their  volumes. 

598.  Examples.  1.  A  rectangle  is  8  in.  long  and  6  in. 
broad.  Find  the  length  and  the  area  of  a  similar  rectangle 
whose  breadth  is  9  in. 

Solution.     6 : 9  =  8  in. :  required  length. 
Therefore,  the  required  length   =  12  in. 
The  area  of  the  given  rectangle  =  48  sq.  in. 
Hence,  48  sq.  in. : required  area  =  62  :92  =  4  :9. 
Therefore,  the  required  area      =  108  sq.  in. 

2.  The  altitude  of  a  right  prism  that  contains  8  cu.  ft.  is 
3  ft.  Find  the  altitude  of  a  similar  right  prism  that  con- 
tains 27  cu.  ft. 

Solution.     3  ft.:  required  altitude  =  V8  :  V27  =  2:3. 
Hence,  the  required  altitude  =  4£  ft. 

Exercise  143. 

1.  If  the  diameter  of  the  moon  is  reckoned  at  2000  mi., 
and  that  of  the  earth  at  8000'  mi.,  find  the  ratio  of  their 
surfaces  and  the  ratio  of  their  volumes. 

2.  If  the  diameters  of  two  circles  are  20  in.  and  40  in., 
find  the  ratio  of  their  circumferences,  and  of  their  surfaces. 

3.  If  the  areas  of  two  circles  are  8000  sq.  in.  and  36,000 
sq.  in.,  respectively,  find  the  ratio  of  their  diameters. 


MENSURATION.  331 

4.  If  the  volumes  of  two  spheres  are  100  cu.  in.  and 
1000  cu.  in.,  respectively,  find  the  ratio  of  their  diameters. 

5.  If  an  ox  7  ft.  in  girth  weighs  1500  lb.,  what  will  be 
the  girth  of  a  similar  ox  that  weighs  2500  lb.? 

6.  The  surface  of  a  pyramid  is  560  sq.  in.  What  is 
the  surface  of  a  similar  pyramid  whose  volume  is  27  times 
as  great  ? 

7.  The  volume  of  a  pyramid  is  1331  cu.  in.  What  is 
the  volume  of  a  similar  pyramid  whose  surface  is  4  times 
as  great  ? 

8.  If  a  well-proportioned  man  5  ft.  10  in.  high  weighs 
160  lb.,  what  should  a  man  6  ft.  high  weigh,  to  the  nearest 
tenth  of  a  pound  ?  What  should  be  the  height,  to  the 
nearest  tenth  of  an  inch,  of  a  man  who  weighs  210  lb.? 

9.  A  three-gallon  jug  and  a  one-gallon  jug  are  similar. 
Find  to  three  decimals  the  ratio  of  their  diameters. 

10.  Two  hills  have  exactly  the  same  shape  ;  one  is 
900  ft.  high,  the  other  1200  ft.  Find  the  ratio  of  their 
surfaces,  and  also  the  ratio  of  their  volumes. 

11.  A  ball  3  in.  in  diameter  weighs  4  lb. ;  another  ball 
of  the  same  metal  weighs  9  lb.  Find  the  diameter  of  the 
second  ball  to  the  nearest  thousandth  of  an  inch. 

12.  If  Apollo's  altar  were  a  perfect  cube  10  ft.  on  an 
edge, -what  would  be  the  edge  of  a  new  cubical  altar  con- 
taining twice  as  much  stone  ? 

13.  A  man  standing  40  ft.  from  a  building  24  ft.  wide 
observed  that,  when  he  closed  one  eye,  the  width  of  the 
building  hid  from  view  90  rods  of  fence  which  was  parallel 
to  the  width  of  the  building.  Find  the  distance  from  the 
eye  of  the  observer  to  the  fence. 

14.  A  bushel  measure  and  a  peck  measure  are  of  the 
same  shape.     Find  the  ratio  of  their  heights. 

15.  If  the  height  and  the  diameter  of  a  cylinder  are 
both  doubled,  in  what  ratio  is  the  volume  altered  ? 


CHAPTER  XVII. 

CONTINUED  FRACTIONS  AND  SCALES  OF  NOTATION. 

599.  As  in  decimals  we  often  require  a  result  accurate 
to  a  specified  number  of  places,  so  in  common  fractions  we 
often  require  the  most  nearly  accurate  value  of  a  ratio 
that  can  be  expressed  by  a  fraction  with  a  denominator 
limited  to  a  certain  size. 

600.  Example.  Find  the  most  nearly  accurate  value  of 
the  ratio  of  the  circumference  of  a  circle  to  the  diameter 
expressed  by  a  fraction  with  a  denominator  less  than  10 ; 
less  than  100  ;   less  than  1000. 

Solution.     The  ratio  3.1416  is  true  to  the  nearest  ten-thousandth. 
Reducing  Jfflfo  to  its  lowest  terms,  we  have  J£fa. 

Then,  as  in  the  margin,  we  divide  the  denom- 
177)1250(7  inator  by  the  numerator;  the  last  divisor  by 

1239  the  last  remainder ;  and  so  on,  as  in  finding 

ll)l77(16  the  greatest  common  measure. 

176  If,  therefore,  we  divide  both  terms  of  the 

'"  fraction  Jjfv  by  the  numerator,  we  have  r-—r  ; 

11  /T77 

and  if  we  omit  the  fraction  in  the  denomi- 
nator, we  have  for  the  required  ratio  with  a  denominator  less  than 
10,  37,  or  y. 

If  we  put  the  fraction  -^  in  the  form  of  tttt  and  omit  the  fraction 

in  the  denominator,  the  ratio  becomes 

3^  =  3^  =  111; 

which  shows  that  272-  is  the  most  nearly  accurate  value  of  the  ratio 
expressed  by  a  fraction  with  a  denominator  less  than  100,  and  that 
fff  is  the  most  nearly  accurate  value  of  the  ratio  expressed  by  a  frac- 
tion with  a  denominator  less  than  1000. 


CONTINUED    FRACTIONS.  333 

601.  Continued  Fractions.  After  the  quotients  have 
been  found  the  results  may  be  written  in  a  fractional  form 
as  follows  : 

1 


7-f 


16+n 


Such  a  fraction  is  called  a  continued  fraction. 

602.  To  find  the  successive  approximate  values  of  a  con- 
tinued fraction  we  begin  at  the  top  and  take  first  one,  then 
two,  then  three,  and  so  on,  of  its  parts.     Thus, 

The  first  approximate  value  is  3. 

The  second  is  3  +  \  =  -2T2-. 

The  third  is  3  +  — L-  =  3  +  T\%  =  f  f  f. 

The  fourth  is  3  + 


7  + 


»+A 


and  this  =  3  +  -\T=  3  +  tf&  =  fffi,  or  3.1416. 

'T7T    , 

603.  In  reducing  the  part  of  a  continued  fraction 
selected  for  an  approximate  value,  we  begin  with  the 
last  fraction. 

Thus,  find  the  value  of  the  continued  fraction 

1 

2  +  -1 


3  +  -L 


-I 


1=     ..     JL-«.     J-=     6, 

4i      3"  3 A      «'  2H      T"- 


334  CONTINUED   FRACTIONS. 


Exercise  144. 

1.  Change  fT,  }£,  f£7,  -\\^  to  continued  fractions. 

2.  Find  the  approximate  values  of  f ^ ;    |^  ;    |f^. 

3.  Find  a  series  of  fractions  approximating  to  0.236  ; 
0.2361;   1.609. 

4.  Find  a  series  of  fractions  approximating  to  0.382  ; 
1.732;   0.6253. 

5.  Find  approximate  values  of  ££}  ;    f  J^ ;   $}£  ;  f  ^£. 

6.  Find  the  proper  fraction  that,  when  changed  to  a 
continued  fraction,  will  have  2,  3,  5,  6,  7  as  quotients. 

7.  Find  a  series  of  fractions  approximating  to  the  ratio 
of  the  pound  troy  (5760  gr.)  to  the  pound  avoirdupois 
(7000  gr.). 

8.  Find  a  series  of  fractions  approximating  to  the  ratio 
of  the  side  of  a  square  to  its  diagonal ;  that  ratio  being 
1:1.414214,  nearly. 

9.  Find  a  series  of  fractions  approximating  to  the  ratio 
of  the  ar  to  the  square  chain,  from  the  equality 

1  ar  =  0.2471  sq.  ch. 

10.  Find  a  series  of  fractions  approximating  to  the  ratio 
of  the  weight  of  the  48-pound  shot  to  the  weight  of  the 
French  shot  of  24k*. 

11.  If  the  mean  diameter  of  the  Earth  is  reckoned  at 
7912  mi.,  and  that  of  Mars  4189  mi.,  find  a  series  of  frac- 
tions approximating  to  the  ratio  of  the  mean  diameters  of 
these  two  planets. 

12.  Find  a  series  of  fractions  approximating  to  the  ratio 
of  a  cubic  yard  to  a  cubic  meter,  from  the  equality 

1  cu.  yd.  =  0.76453cbm. 

13.  Find  a  series  of  fractions  approximating  to  the  ratio 
of  the  kilometer  to  the  mile,  from  the  equality 

1™  =  1.09362  yd. 


SCALES   OF   NOTATION.  335 

14.  Find  the  proper  fraction  that,  if  changed  to  a  con- 
tinued fraction,  will  have  as  quotients  1,  7,  5,  2. 

15.  Find  a  series  of  fractions  approximating  to  0.5236  ; 
approximating  to  0.7854. 

16.  Find  a  series  of  fractions  approximating  to  the  con- 
tinued fraction  that  has  as  quotients  7,  2,  1,  2,  6,  4  ;  that 
has  as  quotients  1,  2,  3,  4,  5,  6. 

Scales  of  Notation. 

604.  The  common  mode  of  representing  numbers  is 
called  the  common  scale  of  notation,  and  io  is  called  its 
radix  or  base. 

605.  In  the  common  or  decimal  scale  every  figure  placed 
to  the  left  of  another  represents  ten  times  as  much  as  if  it 
were  in  the  place  of  that  other. 

606.  Instead  of  the  radix  number  10,  any  other  integral 
number  might  be  used  as  the  base  of  a  system  of  notation. 

Thus,  the  number  6532  stands  for : 

In  the  scale  of  10,  6  X  103  +  5  X  102  +  3  X  10  +  2. 
In  the  scale  of  8,  6  X  83  +  5  x  8a  +  3  X  8  +  2. 
In  the  scale  of    7,     6  X    73  +  5  X    72  +  3  x    7  +  2. 

607.  A  given  number  can  be  changed  from  one  scale  to 
another  scale. 

608.  Examples.     1.   Express  6532  in  the  scale  of  6. 

Solution.  The  quotients  and  remainders  of  the  successive  divi- 
sions by  6  are  as  follows : 


6532 


1088  remainder  4. 

6 1 181   remainder  2. 

6  [30  remainder  1. 

5  remainder  0. 

Therefore,  6532  expressed  in  the  scale  of  6  is  50,124. 


336  SCALES   OF   NOTATION. 

2.  Change  50,124  from  the  scale  of  6  to  the  scale  of  8. 

Solution.  8)50124 

8)3440  remainder  4. 

8)250  remainder  0. 

8  [20  remainder  6. 

1   remainder  4. 

Therefore,  the  number  required  is  14,604. 

Since  50,124  is  in  the  scale  of  6,  each  figure  has  six  times  the  value  it 
would  have  one  place  to  the  right.  Hence,  at  the  beginning  we  have 
to  divide  6  X  5  +  0  by  8,  and  we  get  3  for  the  quotient  and  6  for  the 
remainder.  The  next  partial  dividend  is  6  X  6  4-  1,  or  37,  and  this 
divided  by  8  gives  4  for  the  quotient  and  5  for  the  remainder.  The 
next  partial  dividend  is  6  X  5  +  2,  or  32,  and  this  divided  by  8  gives 
4  for  the  quotient  and  0  for  the  remainder ;  and  so  on. 

3.  Change  14,604  from  the  scale  of  8  to  the  scale  of  10. 

Solution.  10114604 

10]  1215  remainder  2. 
10  [101   remainder  3. 
6  remainder  5. 
Therefore,  the  required  number  is  6532. 

4.  Add  56,432  and  15,646  (scale  of  7). 

Solution.  The  process  differs  from  that  in  the  decimal 
scale  only  in  that  when  a  sum  greater  than  seven  is  reached, 
we  divide  by  seven  (not  ten),  set  down  the  remainder,  and 


add  the  quotient  with  the  next  column. 

5.  Subtract  34,561  from  61,235  (scale  of  8). 

Solution.     When  the  number  of  any  order  of  units  in 

the  minuend  is  less  than  the  number  of  the  corresponding 

order  in  the  subtrahend,  we  increase  the  number  in  the 
24454 

minuend  by  eight  instead  of  ten  as  in  the  common  scale. 

6.  Multiply  5732  by  428  (scale  of  9). 

5732 

Solution.     We  divide  each  partial  product  by  nine,  set 

down  the  remainder,  and  add  the  quotient  to  the  next 

partial  product. 
8o28o 

2712127 


SCALES   OF   NOTATION.  337 

Divide  2,712,127  by  5732  (scale  of  9). 

428 
5732)2712127 
25238 
17722 
12564 


51477 
51477 


The  operations  of  multiplication  and  subtraction  involved  in  this 
problem  are  precisely  the  same  as  in  the  decimal  notation.  The  only 
difference  is  that  the  radix  number  is  9  instead  of  10. 

Exercise  145. 

Change  4852  of  the  common  scale  to  : 

1.  The  scale  of  7.  5.    The  scale  of  6. 

2.  The  scale  of  2.  6.   The  scale  of  5. 

3.  The  scale  of  9.  7.    The  scale  of  8. 

4.  The  scale  of  3.  8.   The  scale  of  4. 

Change : 

9.  54,231  of  the  scale  of  6  to  the  common  scale. 

10.  54,231  of  the  scale  of  7  to  the  common  scale. 

1 1 .  54,231  of  the  scale  of  8  to  the  common  scale. 

1 2.  54,231  of  the  scale  of  9  to  the  common  scale. 

Perform  the  following  arithmetical  processes  : 

13.  Add  67,814;   76,406;   88,718  (scale  of  9). 

14.  Add  44,231 ;   13,432  ;    12,304  (scale  of  5). 

15.  Subtract  77,614  from  114,672  (scale  of  8). 

16.  Subtract  52,515  from  112,252  (scale  of  6). 

17.  Multiply  14,612  by  6502  (scale  of  7). 

18.  Multiply  72,645  by  46,723  (scale  of  8). 

19.  Divide  162,542  by  6522  (scale  of  7). 

20.  Divide  468,722  by  5432  (scale  of  9). 


CHAPTER   XVIII. 

SERIES. 

609.  Series.  A  succession  of  numbers  that  proceed 
according  to  some  fixed  law  is  called  a  series.  The  succes- 
sive numbers  are  called  the  terms  of  the  series. 

610.  A  series  that  ends  at  some  particular  term  is  called 
a  finite  series.  A  series  that  continues  without  end  is 
called  an  infinite  series. 

611.  The  number  of  different  kinds  of  series  is  un- 
limited ;  in  this  chapter  we  shall  consider  only  Arith- 
metical Series,  Geometrical  Series,  and  Harmonical  Series. 

Arithmetical  Progression. 

612.  A  series  of  numbers  that  increase  or  decrease  by  a 
common  difference  is  called  an  Arithmetical  Series  or  an 
Arithmetical  Progression. 

Thus,  the  numbers  6,  8,  11,  14  form  an  arithmetical  progression 
with  a  common  difference  3 ;  and  the  numbers  12,  10,  8,  6  form  an 
arithmetical  progression  with  a  common  difference  2. 

613.  In  the  increasing  arithmetical  progression 
1st  2d  3d  4th  6th  6th 
2,             5,             8,             11,             14,  17, 

we  find  any  term,  as  the  6th,  by  adding  to  the  first  term 
the  product  of  the  common  difference  by  a  number  one 
less  than  the  number  of  the  term  :  2  +  (3  X  5),  or  17. 
In  the  decreasing  arithmetical  progression 

1st     2d     3d     4th    5th    6th     7th 
50,    46,    42,    38,    34,    30,    26, 


SERIES.  339 

we  find  any  term,  as  the  7th,  by  subtracting  from  the  first 
term  the  product  of  the  common  difference  by  a  number 
one  less  than  the  number  of  the  term  :  50  —  (4  X  6),  or  26. 
Hence, 

614.  To  Find  Any  Term  of  an  Arithmetical  Progression, 

Multiply  the  common  difference  by  a  number  one  less  than 
the  number  of  the  required  term.  Add  this  product  to  the 
first  term  if  the  series  is  an  increasing  series ;  subtract  this 
product  from  the  first  term  if  the  series  is  a  decreasing  series. 

Exercise  146. 

1.  Find  the  seventh  term  of  the  series  3,  5,  7,  etc. 

2.  Find  the  fifteenth  term  of  the  series  2,  7,  12,  etc. 

3.  Find  the  sixth  term  of  the  series  2,  2|,  3f,  etc. 

4.  Find  the  twentieth  term  of  the  series  2,  3  J,  4^,  etc. 

5.  Find  the  seventh  term  of  the  series  21,  19,  17,  etc. 

6.  Find  the  twelfth  term  of  the  series  18,  17  J,  16f,  etc. 

7.  If  the  first  term  of  a  series  is  5,  and  the  common 
difference  2j,  find  the  thirteenth  and  eighteenth  terms. 

8.  If  the  fourth  term  of  a  series  is  18,  and  the  common 
difference  3,  find  the  seventh  and  eleventh  terms. 

9.  If  the  fifth  term  of  a  decreasing  series  is  52,  and  the 
common  difference  3^,  find  the  twelfth  and  eighteenth  terms. 

10.  If  the  fourth  term  of  a  series  is  14,  and  the  twelfth 
term  38,  what  is  the  common  difference  ? 

Hint.  The  difference  between  the  fourth  and  twelfth  terms  is 
evidently  eight  times  the  common  difference. 

Find  the  common  difference  in  a  series  : 

11.  If  the  fourth  term  is  12  and  the  seventh  term  27. 

12.  If  the  first  term  is  20  and  the  fourth  term  40. 

13.  If  the  first  term  is  2  and  the  eleventh  term  20. 

14.  If  the  third  term  is  7  and  the  eighth  term  12^. 

15.  If  the  first  term  is  1  and  the  fourth  term  19. 


340  SERIES. 

615.  The  sum  of  seven  terms  of  the  series  3,  5,  7,  etc.,  is 

3+  5  +  7+  9  +  11  +  13  +  15, 
in  reverse  order  is  15  +  13  +  11+    9+    7+   5+    3. 

Hence,  twice  the  sum  is  18  + 18  + 18  + 18  +  18  + 18  + 18, 
or  7  X  18. 

Therefore,  the  sum  is  £  (7  X  18),  or  63. 

Here  7  is  the  number  of  terms,  and  18  is  the  sum  of  the 
first  and  last  terms.     Hence, 

616.  To  Find  the  Sum  of  the  Terms  of  an  Arith- 
metical Progression, 

Multiply  one  half  the  sum  of  the  first  and  last  terms  by 
the  number  of  terms. 

Thus,  the  sum  of  eight  terms  of  the  series  whose  first  term  is  3  and 
last  term  38  is  8  X  £  (3  +  38)  =  164. 

Exercise  147. 

1.  Find  the  sum  of  1,  5,  9,  etc.,  to  twenty  terms. 

2.  Find  the  sum  of  4,  5£,  7,  etc.,  to  eight  terms. 

3.  Find  the  sum  of  8,  7f ,  7£,  etc.,  to  sixteen  terms. 

4.  Find  the  sum  of  20,  18£,  16£,  etc.,  to  seven  terms. 

5.  Find  the  sum  of  the  first  twenty  natural  numbers. 

6.  Find  the  sum  of  the  natural  numbers  from  37  to  53 
both  inclusive. 

7.  Find  the  sum  of  a  series  of  thirty  terms,  if  the  first 
term  is  21  and  the  last  59. 

8.  Find  the  sum  of  the  series  whose  first  two  terms  are 
3  and  9,  and  the  last  term  75. 

9.  Find  the  sum  of  a  series  of  twenty  terms  whose 
third  and  fifth  terms  are  10  and  15,  respectively. 

10.  A  body  falls  through  a  space  of  16^  ft.  in  the  first 
second  of  its  fall,  and  in  each  succeeding  second  32£  ft. 
more  than  in  the  second  just  before.  How  far  will  a  stone 
fall  in  the  seventh  second  ?     How  far  in  seven  seconds  ? 


SERIES.  341 

11.  A  travels  8  miles  the  first  day,  11  miles  the  second, 
14  miles  the  third,  and  so  on,  and  overtakes  in  17  days  B 
who  started  at  the  same  time,  and  traveled  at  a  uniform 
rate.     What  is  B's  rate  per  day  ? 

12.  In  a  potato  race  100  potatoes  are  placed  in  a  straight 
line  3  ft.  distant  from  each  other.  A  boy,  starting  from  a 
basket  3  ft.  from  the  first  potato,  is  required  to  pick  them 
up  one  by  one  and  carry  them  to  the  basket.  To  finish  the 
race,  how  far  must  the  boy  run  ? 

13.  How  many  times  a  day  does  a  clock  strike  that 
strikes  the  hours  only  ? 

14.  A  body  falls  through  a  space  of  4.9m  in  the  first 
second  of  its  fall,  and  in  each  succeeding  second  9.8m  more 
than  in  the  second  just  before.  A  stone  dropped  from  a 
balloon  was  35  seconds  in  reaching  the  ground.  How  high 
was  the  balloon  ? 

Geometrical  Progression. 

617.  A  series  of  numbers  each  term  of  which  after 
the  first  is  obtained  by  multiplying  the  preceding  term 
by  a  constant  multiplier  is  called  a  Geometrical  Series, 
or  a  Geometrical  Progression.  The  constant  multiplier 
is  called  the  ratio. 

Thus,  3,  6,  12,  24  form  a  geometrical  progression  with  a  ratio  2 ; 
and  97  3,  1,  -J-  form  a  geometrical  progression  with  a  ratio  £. 

618.  In  the  geometrical  progression 

1st  2d  3d  4th  5th  6th 

2,         6,         18,         54,         162,         486, 

the  second  term  is  2  X  3;  the  third  term  is  2  X  32;  the  fourth 
term  is  2  X  33;  the  fifth  term  is  2  X  34;  and  so  on.     Hence, 

619.  To  Find  Any  Term  of  a  Geometrical  Progression, 

Multiply  the  first  term  by  that  power  of  the  ratio  that  is 
one  less  than  the  number  of  the  term  required. 


342  SERIES. 

620.  If  any  two  consecutive  terms  of  a  geometrical  pro- 
gression are  known,  the  ratio  may  be  found  by  dividing 
the  second  of  these  terms  by  the  first. 

If  two  terms  not  consecutive  are  known,  the  ratio  may 
be  found  as  in  the  following 

Example.  Find  the  ratio  in  a  geometrical  progression  if 
the  second  and  sixth  terms  are  5  and  80,  respectively. 

Solution.  6th  term  =  2d  term  x  (ratio)4. 

Therefore,  (ratio)4  =  4£  =  16, 

and  ratio  =  Vl6  =  2. 

Exercise  148. 

1.  Find  the  eighth  term  of  the  series  2,  6,  18,  etc. 

2.  Find  the  fifth  term  of  the  series  8,  4,  2,  etc. 

3.  Find  the  seventh  term  of  the  series  2,  3,  4^,  etc. 

4.  Find  the  sixth  term  of  the  series  4,  2§,  1$,  etc. 

5.  Find  the  eighth  term  of  the  series  4,  10,  25,  etc. 

6.  Find  the  fifth  term  of  the  series  J,  ^  gij,  etc. 

7.  Find  the  ninth  term  of  the  series  4,  2,  1,  etc. 

8.  Find  the  sixth  term  of  the  series  6,  9,  13£,  etc. 

9.  Write  the  first  six  terms  of  the  geometrical  series 
whose  fifth  and  sixth  terms  are  112  and  224,  respectively. 

10.  The  seventh  and  ninth  terms  of  a  geometrical  series 
are  100  and  144,  respectively.     Find  the  twelfth  term. 

11.  A  capital  of  $1000  is  increased  by  ^  of  itself  each 
year.     What  will  it  be  at  the  beginning  of  the  fifth  year  ? 

12.  A  capital  of  $1000  is  increased  by  jfo  of  itself  each 
year.     What  will  it  be  at  the  beginning  of  the  sixth  year  ? 

621.  In  the  geometrical  progression  4,  12,  36,  108,  324, 
etc.,  the  sum  of  five  terms  is  4  +  12  +  36  +  108  +  324. 

If  we  multiply  this  sum  by  the  ratio  3,  and  from  the 
product  subtract  the  sum  of  the  five  terms,  we  shall  have 
twice  the  sum  of  the  terms.     That  is, 


SERIES.  343 


12  +  36  +  108  +  324  +  972 
4  +  12  +  36  +  108  +  324 


Therefore,  the  sum  = 


972-4. 
972-4 


The  numerator  is  the  difference  between  the  product  of 
the  last  term  by  the  ratio  and  the  first  term  ;  and  the 
denominator  is  the  ratio  minus  1.     Hence, 

622.  To  Find  the  Sum  of  the  Terms  of  a  Geometrical 
Progression, 

Multiply  the  last  term  by  the  ratio,  and  subtract  the  first 
term  from  the  product.  Divide  the  remainder  by  the  ratio 
minus  1. 

If  the  ratio  is  less  than  1, 

Multiply  the  last  term  by  the  ratio,  and  subtract  the  product 
from  the  first  term.  Divide  the  remainder  by  1  minus  the 
ratio. 

Exercise  149. 

1.  Find  the  sum  of  2,  6,  18,  etc.,  to  six  terms. 

2.  Find  the  sum  of  1,  2,  4,  etc.,  to  nine  terms. 

3.  Find  the  sum  of  3,  9,  27,  etc.,  to  five  terms. 

4.  Find  the  sum  of  2,  3,  4£,  etc.,  to  eight  terms. 

5.  Find  the  sum  of  1,  J,  ^,  etc.,  to  eight  terms. 

6.  Find  the  sum  of  1,  ^,  i,  etc.,  to  ten  terms. 

7.  Find  the  sum  of  £,  £,  },  etc.,  to  eight  terms. 

8.  Find  the  sum  of  the  first  six  terms  of  the  series 
whose  first  term  is  3  and  ratio  5. 

9.  Find  the  sum  of  the  first  eight  terms  of  the  series 
whose  first  term  is  3  and  ratio  ^. 

10.  A  man  saved  in  one  year  $64,  and  in  each  succeed- 
ing year,  for  9  years  more,  l£  times  as  much  as  in  the  pre- 
ceding year.     Find  the  whole  amount  he  saved. 


344  SERIES. 

623.  If  we  represent  the  first  term  by  a,  the  last  term 
by  I,  the  ratio  by  r,  the  number  of  terms  by  n,  and  the  sum 
of  the  terms  by  s,  we  have,  if  the  ratio  is  less  than  1  (§  622), 

a  —  r  X  I 

Now,  since  Z=  a  X  r"_1,  rl  =  ar";   and  we  have 
_  a  —  ar*  _  a  X  (1  —  r") 
1  —  r  1  —  r 

Since  r  is  less  than  1,  r"  becomes  smaller  as  n  becomes 
larger ;  and  when  n  is  too  large  to  be  counted,  r"  becomes 
too  small  to  be  considered.     We  then  have 

s  = Hence, 

1  —  r 

624.  To  Find  the  Sum  of  the  Terms  of  an  Infinite, 
Decreasing  Geometrical  Progression, 

Divide  the  first  term  by  1  minus  the  ratio. 

Exercise  150. 
Find  the  sum  of  the  infinite  series  : 

1.    h  h  h  •  6-   0.212121 


2.  §,  |,  A, •  7.  0.9999 

3.  i,A,A> '  8'  0.232323- 

4.  h  A,  t**>  •  9-  0.36848484 

5.  0.171717 10.  0.15272727 


625.  A  series  is  called  a  Harmonical  Series,  or  a  Har- 
monical  Progression,  if  the  reciprocals  of  its  terms  form  an 
arithmetical  series. 

Thus,  the  numbers  1,  i,  i,  £  form  a  harmonical  progression,  since 
1,  2,  3,  4,  the  reciprocals  of  the  terms,  are  in  arithmetical  progression. 

626.  Questions  involving  a  harmonical  progression  may 
be  solved  by  writing  the  reciprocals  of  the  terms  so  as  to 
form  an  arithmetical  progression. 


CHAPTER  XIX. 

COMMON  LOGARITHMS. 

627.  In  the  common  system  of  notation  the  expression 
of  numbers  is  founded  on  their  relation  to  10. 

Thus,  5434  indicates  that  this  number  consists  of  103  five  times, 
102  four  times,  10  three  times,  and  four  units. 

628.  But  any  number  may  be  expressed,  exactly  or 
approximately,  as  a  power  of  10. 

Thus,  5434  is  greater  than  103  and  less  than  104,  and  is  expressed 
approximately  by  103-7351. 

629.  When  a  number  is  expressed  as  a  power  of  10,  the 
exponent  of  10  is  called  the  logarithm  of  that  number. 

Thus,  if  108-7861  =  5434,  the  logarithm  of  5434  =  3.7351,  a  state- 
ment that  is  generally  written  log  5434  =  3.7351. 

.    .  .        102 

630.  By  division  — 2  =  1  ;   and  by  the  rule  for  expo- 

102 
nents  (§  108)  — 2  ==  102~2  =  10°.     Therefore,  10°  =  1. 

Since        10°=        1,  log        1  =  0, 

10x  =      10,  log      10  =  1, 

102  =    100,  log    100  =  2, 

103  =  1000,  log  1000  =  3,  and  so  on. 

631.  It  is  often  convenient  to  write  the  reciprocals  of 
powers  of  10  in  an  integral  form;   —  is  written  10_1;   —* 

is  written  10  _2 ;    — -„  is  written  10  ~3 ;    and  so  on. 
103 

Note.  An  exponent  with  the  minus  sign  prefixed  is  called  a  nega- 
tive exponent ;  and  an  exponent  with  the  plus  sign  prefixed,  or  with- 
out any  sign,  is  called  a  positive  exponent. 


0  and 

1, 

1  and 

o 

2  and 

3, 

0  and  ■ 

-1, 

1  and 

-2, 

2  and- 

-3, 

and 

so  on. 

346  COMMON    LOGARITHMS. 

632.  Since  lO"1  =  0.1,  log  0.1      =  —  1, 

10-*  =  0.01,  log  0.01    =  —  2, 

10-3  =  0.001,        log  0.001  =  —  3,  and  so  on. 

633.  It  is  evident,  therefore,  that  the  logarithms  of  all 
numbers  between 

1  and       10  will  lie  between 

10  and  100  will  lie  between 
100  and  1000  will  lie  between 

1  and  0.1  will  lie  between 
0.1  and  0.01  will  lie  between 
0.01  and  0.001  will  lie  between 

634.  If  a  number  is  less  than  1,  its  logarithm  is  nega- 
tive (§  633),  but  is  written  in  such  a  form  that  its  decimal 
part  is  always  positive. 

Thus,  log  0.0284  =  -  (1.5467)  =  (—  1)+  (—  .5467)  =  -  2  +  .4533. 

If  the  integral  part  of  a  logarithm  is  negative,  the  minus 
sign  is  written  over  the  integral  part. 
Thus,  log  0.0284  is  written  2.4533. 

635.  Every  logarithm,  therefore,  consists  of  two  parts  : 
a  positive  or  negative  integral  number  called  the  character- 
istic, and  a  positive  decimal  called  the  mantissa. 

Thus,  in  the  logarithm  2.4533,  the  integral  number  2  is  the  charac- 
teristic, and  the  decimal  .4588  is  the  mantissa.  In  the  logarithm 
2.  1638,  the  negative  integral  number  —  2  is  the  characteristic  and  the 
positive  decimal  .4533  is  the  mantissa. 

636.  If  a  logarithm  has  a  negative  characteristic,  it  is 
customary  to  change  its  form  by  adding  10  or  a  multiple 
of  10  to  the  characteristic,  and  indicating  the  subtraction 
of  the  same  number  from  the  result. 

Thus,  the  logarithm  2.4533  is  written  8.4533  —  10,  and  the  loga- 
rithm 13.4533  is  written  7.4533  -  20. 


COMMON    LOGARITHMS.  34? 

637.  From  an  inspection  of  §  633,  we  deduce  the  follow- 
ing rules  for  writing  the  characteristic  of  a  logarithm  : 

Rule  1.  If  the  given  number  is  greater  than  1,  make  the 
characteristic  of  its  logarithm  one  less  than  the  number  of 
figures  to  the  left  of  the  decimal  point  in  the  number. 

Rule  2.  If  the  given  number  is  less  than  1,  make  the 
characteristic  of  its  logarithm  negative,  and  one  more  than 
the  number  of  zeros  between  the  decimal  point  and  the  first 
significant  figure  of  the  given  number. 

Thus,  the  characteristic  of  log  7849.27  is  3 ;  the  characteristic  of 
log  0.037  is  -  2,  or  8.0000  -  10. 

638.  The  mantissa  of  the  logarithm  of  any  number 
depends  only  upon  the  sequence  of  the  digits  of  the 
number,  and  is  unchanged  so  long  as  the  sequence  of 
the  digits  remains  the  same. 

For  changing  the  position  of  the  decimal  point  in  a  number  is 
equivalent  to  multiplying  or  dividing  the  number  by  a  power  of  10. 
Its  logarithm,  therefore,  will  be  increased  by  or  diminished  by  the 
exponent  of  that  power  of  10  ;  and  since  this  exponent  is  integral,  the 
mantissa,  or  decimal  part  of  the  logarithm,  will  be  unchanged. 
Thus,         27,196  =  104-434*,  2.7196  =  lO0-434*, 

2719.6  =  103-434*,  0.27196  =  lO9-4345-™, 

27.196  =  101-4345,  0.0027196  =  W.ms-w, 

639.  A  Four-place  Table  of  Logarithms.  A  four-place 
table  of  logarithms  is  given  on  pages  350  and  351.  This 
table  contains  the  mantissas  of  the  logarithms  of  all  inte- 
gral numbers  under  1000,  the  decimal  point  being  omitted. 

Note.  Tables  containing  logarithms  to  more  decimal  places  can 
be  procured,  but  this  table  will  serve  for  many  practical  uses,  and  will 
enable  the  student  to  understand  the  use  of  five-place,  seven-place,  and 
ten-place  logarithms  in  work  that  requires  greater  accuracy. 

640.  In  working  with  a  four-place  table,  numbers  cor- 
responding to  logarithms  will  be  correct  to  four  significant 
digits. 


348  COMMON    LOGARITHMS. 

To  Find  the  Logarithm  of  a  Given  Number. 

641.  The  characteristic  of  the  logarithm  is  determined 
by  the  rules  of  §  637. 

642.  If  the  given  number  consists  of  a  single  digit,  as  4, 
8,  etc.,  the  mantissa  of  its  logarithm  is  the  same  as  the 
mantissa  of  the  logarithm  of  40,  80,  etc. 

643.  If  the  given  number  contains  two  digits,  it  is  in 
the  column  headed  N,  and  the  mantissa  of  its  logarithm  is 
on  the  same  line,  and  in  the  column  headed  0. 

Thus,  log  28  =  1.4472,  log  0.086  =  8.9345  -  10, 

log  40  =  1.6021,  log  7         =  0.8451. 

644.  If  the  given  number  contains  three  digits,  or  three 
significant  digits  followed  by  one  or  more  zeros,  the  first 
two  digits  of  the  number  are  in  the  column  headed  N,  and 
the  third  at  the  top  of  the  page  in  the  line  containing  the 
figures  0,  1,  2,  etc.  The  mantissa  of  its  logarithm  is  in 
the  column  headed  by  the  third  figure  and  on  the  same  line 
with  the  first  two  figures. 

Thus,        log  742    =2.8704,  log  84,100    =4.9248, 

log  6090  =  3.7846,  log  0.00261  =  7.4166  -  10. 

645.  If  the  given  number  contains  four  or  more  digits, 
the  mantissa  of  its  logarithm  is  found  as  in  the  following 

Examples.     1.    Find  the  logarithm  of  2034. 

Solution.  The  required  mantissa  is  (§  638)  the  same  as  the  man- 
tissa for  203.4  ;  hence,  it  is  found  by  adding  to  the  mantissa  for  203 
four  tenths  of  the  difference  between  the  mantissas  for  203  and  204. 

The  mantissa  for  203  is  3076 ;   and  for  204  is  3096. 

The  difference  between  the  mantissas  for  203  and  204  is  21,  and  0.4 
of  21  =  8.     Hence,  8  must  be  added  to  3075. 

Therefore,  the  mantissa  for  203.4  is  3075  +  8  =  3083. 

Therefore,  log  2034  =  3.3083. 

2.    Find  the  logarithm  of  0.0015764. 

Solution.  The  required  mantissa  is  (§  638)  the  same  as  the  man- 
tissa for  157.64  ;  hence,  we  add  to  the  mantissa  for  157  sixty-four 


COMMON   LOGARITHMS.  349 

hundredths  of  the  difference  between  the  mantissas  for  157  and 
158. 

The  mantissa  for  157  is  1959  j   and  for  158  is  1987. 

The  difference  between  the  mantissas  for  157  and  158  is  28,  and 
0.64  of  28  =  18.     Hence,  18  must  be  added  to  1959. 

Therefore,  the  mantissa  for  157.64  is  1959  +  18  =  1977. 

Therefore,  log  0.0015764  =  7.1977  -  10. 

Note.  When  the  fraction  of  a  unit  in  the  part  to  be  added  to  the 
mantissa  for  three  figures  is  less  than  0.5,  it  is  neglected ;  when  it  is 
0.5  or  more,  it  is  taken  as  one  unit. 

Exercise  151. 
Find  the  logarithm  of  : 


1. 

70. 

6. 

6897. 

11. 

77,860. 

16. 

5.0009. 

2. 

101. 

7. 

9901. 

12. 

30,127. 

17. 

0.3769. 

3. 

333. 

8. 

4389. 

13. 

730.84. 

18. 

0.070707. 

4. 

3491. 

9. 

1111. 

14. 

0.008765. 

19. 

0.03723. 

5. 

1866. 

10. 

58,343. 

15. 

8.0808. 

20. 

98.871. 

646.  From  §  633,  we  deduce  the  following  rules  for  writ- 
ing the  decimal  point  in  the  antilogarithm  of  a  logarithm  ; 
that  is,  in  the  number  corresponding  to  a  logarithm  : 

Rule  1.  If  the  characteristic  of  the  given  logarithm  is 
positive,  make  the  number  of  figures  in  the  integral  part  of 
its  antilogarithm  one  more  than  the  number  of  units  in  the 
characteristic  of  the  logarithm. 

Rule  2.  If  the  characteristic  of  the  given  logarithm  is 
negative,  make  the  number  of  zeros  between  the  decimal  point 
and  the  first  significant  figure  of  its  antilogarithm  one  less 
than  the  number  of  units  in  the  characteristic  of  the  loga- 
rithm. 

Thus,  the  number  corresponding  to  the  logarithm  4.7542  contains 
five  figures  in  its  integral  part ;  the  decimal  corresponding  to  the  log- 
arithm 7.2816  —  10,  that  is  3.2816,  contains  two  zeros  between  the 
decimal  point  and  the  first  significant  figure. 


350 


COMMON    LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 
13 
14 

0000 
0414 
0792 
1139 
1461 

0043 
0453 
0828 
1173 
1492 

0086 
0492 
0864 
1206 
1523 

0128 
0531 
0899 
1239 
1553 

0170 
0569 
0934 
1271 
1584 

0212 
0607 
0969 
1303 
1014 

0253 
0045 
1004 
1335 
1044 

0294 
0682 
1038 
1367 
1673 

0334 
0719 
1072 
1399 
1703 

0374 
0755 
1106 
1430 
1732 

15 

16 
17 
18 
19 

1761 
2041 
2304 
2553 

2788 

1790 
2068 
2330 
2577 
2810 

1818 
2095 
2355 
2601 
2833 

1847 
2122 
2380 
2625 
2856 

1875 
2148 
2405 
2648 
2878 

1903 
2175 
2430 
2672 
2900 

1931 
2201 
2455 
2695 
2923 

1959 
2227 
2480 
2718 
2945 

1987 
2253 
2504 
2742 
2967 

2014 
2279 
2529 
2705 
2989 

20 

21 
22 
23 
24 

3010 
3222 
3424 
3617 
3802 

3032 
3243 
3444 
3636 
3820 

3054 
3263 
3464 
3655 
3838 

3075 
3284 
3483 
8674 

3856 

3096 
3304 
3502 
3692 
3874 

3118 
3324 
3522 
3711 
3892 

3139 
3345 
3541 
8729 
3909 

3160 
3365 
3560 
3747 
3927 

3181 
3385 
3579 
3766 
3945 

3201 
3404 
3598 
3784 
3962 

25 

26 

27 
28 
29 

3979 
4150 
4314 
4472 
4624 

3997 
4166 
4330 

4487 
4639 

4014 
4183 
4346 
4502 
4654 

4031 
4200 
4362 
4518 
4669 

4048 
4216 
4378 
4533 

4683 

4005 
4232 
4393 

4548 
4698 

4082 
4249 
4409 
4564 
4713 

4099 
4265 
4425 
4579 
4728 

4116 
4281 
4440 
4594 
4742 

4133 
4298 

4456 
4009 
4757 

30 
31 
32 
33 
34 

4771 
4914 
5051 
5185 
5315 

4786 
4928 
5065 
5198 
5328 

4800 
4942 
5079 
5211 
5340 

4814 
4955 
6092 
6224 
5353 

4829 
4960 

6105 
6237 
5366 

4843 
4983 
6119 
5250 
5378 

4857 
4997 
6132 
5263 
5391 

4871 
5011 
5145 
5276 
6403 

4886 
5024 
6159 
6289 
5416 

4900 
5038 
5172 
5302 
5428 

35 
36 
37 
38 
39 

5441 
5563 
5682 
5798 
5911 

5453 
6575 
5694 
5809 
6922 

5465 
5587 
6705 
5821 
6933 

5478 
5599 
5717 
6832 
6944 

5490 
6611 
5729 
5843 
5955 

5502 
5623 
5740 
6855 
6966 

5514 
5635 
6752 
5866 
6977 

6527 
6647 

5703 

5877 
6988 

5539 
5658 
5775 
5888 
5599 

5551 
5670 
5786 
5899 
6010 

40 
41 
42 
43 
44 

6021 
6128 
6232 
6335 
6435 

6031 
6138 
6243 
5345 
6444 

6042 
6149 
8258 

6355 
6454 

6053 
6100 
6203 
6305 
6464 

0004 
6170 
6274 
6375 
6474 

6075 
6180 
6284 
6385 
6484 

6085 
6191 
6294 
6395 
6493 

6096 
0201 
6304 
6406 
6003 

6107 
0212 
6314 

0415 
0513 

6117 
6222 
6325 
6425 
6522 

45 

46 
47 
48 
49 

6532 
6628 
6721 
6812 
6902 

6542 
6637 
6730 
6821 
6911 

6551 
6646 
6739 
6830 
6920 

6561 
6656 
6749 
6839 
6928 

6571 
6665 
6758 
6848 
6937 

6580 
6675 
6767 
6857 
6946 

6590 
6684 
6776 
6866 
6955 

6599 
6693 
6785 
6875 
6964 

6609 
6702 
07!  »4 
6884 
6972 

6618 
6712 
6803 
6893 
6981 

50 
51 
52 
53 
54 

6990 
7076 
7160 
7243 
7324 

6998 
7084 
7168 
7251 
7332 

7007 
7093 
7177 
7259 
7340 

7016 
7101 
7185 
7207 
7348 

7024 
7110 
7193 

7275 
7350 

7033 
7118 
7202 
7284 
7364 

7042 
7126 
7210 
7292 
7372 

7050 
7135 
7218 
7300 
7380 

7059 
7143 
7220 
7308 
7388 

7067 
7152 
7235 
7316 
7396 

COMMON   LOGARITHMS. 


851 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

56 
57 
58 
59 

7404 

7482 
7559 
7634 
7709 

7412 
7490 
7566 
7642 
7716 

7419 
7497 
7574 
7649 
7723 

7427 
7505 
7582 
7657 
7731 

7435 
7513 
7589 
7664 
7738 

7443 
7520 
7597 
7672 
7745 

7451 

7528 
7604 

7679 

7752 

7459 
7536 
7612 
7686 
7760 

7466 
7543 
7619 
7694 

7767 

7474 

7551 
7627 
7701 

7774 

60 

61 
62 
63 
64 

7782 
7853 
7924 
7993 
80G2 

7789 
7860 
7931 
8000 
8069 

7796 
7808 
7938 
8007 
8075 

7803 
7875 
7945 
8014 

8082 

7810 
7882 
7952 
8021 

8089 

7818 
7889 
7959 
8028 
8096 

7825 
7896 
7966 
8035 
8102 

7832 
7903 
7973 
8041 
8109 

7839 
7910 
7980 
8048 
8116 

7846 
7917 
7987 
8055 
8122 

65 

66 
67 
68 
69 

8129 
8195 
8261 
8325 
8388 

8136 
8202 
8267 
8331 
8395 

8142 
8209 
8274 
8338 
8401 

8149 
8215 

8280 
8344 
8407 

8156 
8222 
8287 
8351 
8414 

8162 
8228 
8293 
8357 
8420 

8169 
8235 
8299 
8363 
8426 

8176 
8241 
8306 
8370 
8432 

8182 
8248 
8312 
8376 
8439 

8189 
8254 
8319 
8382 
8445 

70 

71 

72 
73 

74 

8451 
8513 
8573 
8633 
8692 

8457 
8519 
8579 
8639 
8698 

8463 
8525 
8585 
8645 
8704 

8470 
8531 
8591 
8651 
8710 

8476 
8537 
8597 
8657 
8716 

8482 
8543 
8603 
8663 

8722 

8488 
8549 
8609 
8669 
8727 

8494 
8555 
8615 
8675 
8733 

8500 
8561 
8621 
8681 
8739 

8506 
8567 
8627 
8686 
8745 

75 

76 

77 
78 
79 

8751 
8808 
8865 
8921 
8976 

8756 
8814 
8871 
8927 
8982 

8762 
8820 
8876 
8932 

8987 

8768 
8825 
8882 
8938 
8993 

8774 
8831 
8887 
8943 
8998 

8779 
8837 
8893 
8949 
9004 

8785 
8842 
8899 
8954 
9009 

8791 

8848 
8904 
8960 
9015 

8797 
8854 
8910 
8965 
9020 

8802 
8859 
8915 
8971 
9025 

80 

81 

82 
83 

84 

9031 
9085 
9138 
9191 
9243 

9036 
9090 
9143 
9196 

9248 

9042 
9096 
9149 
9201 
9253 

9047 
9101 
9154 
9206 
9258 

9053 
9106 
9159 
9212 
9263 

9058 
9112 
9165 
9217 
9269 

9063 
9117 
9170 

9222 
9274 

9069 
9122 
9175 
9227 
9279 

9074 
9128 
9180 
9232 
9284 

9079 
9133 
9186 
9238 
9289 

85  - 

86 
87 
88 
89 

9294 
9345 
9395 
9445 
9494 

9299 
9350 
9400 
9450 
9499 

9304 
9355 
9405 
9455 
9504 

9309 
9360 
9410 
9400 
9509 

9315 
9365 
9415 
9465 
9513 

9320 
9370 
9420 
9469 
9518 

9325 
9375 
9425 
9474 
9523 

9330 
9380 
9430 
9479 
9528 

9335 
9385 
9435 
9484 
9533 

9340 
9390 
9440 
9489 
9538 

90 
91 
92 
93 
94 

9542 
9590 
9638 
9685 
9731 

9547 
9595 
9643 
9689 
9736 

9552 
9600 
9647 
9694 
9741 

9557 
9005 
9652 
9699 
9745 

9562 
9609 
9657 
9703 
9750 

9566 
9614 
9661 
9708 
9754 

9571 
9619 
9666 
9713 
9759 

9576 
9624 
9671 
9717 
9763 

9581 
9628 
9675 
9722 
9768 

9586 
9633 
9680 
9727 
9773 

95 

96 
97 

98 
99 

9777 
9823 
9868 
9912 
9956 

9782 
9827 
9872 
9917 
9961 

9786 
9832 
9877 
9921 
9965 

9791 

9836 
9881 
9926 
9969 

9795 
9841 
9886 
9930 
9974 

9800 
9845 
9890 
9934 

9978 

9805 
9850 
9894 
9939 
9983 

9809 
9854 
9899 
9943 

9987 

9814 
9859 
9903 
9948 
9991 

9818 
9863 
9908 
9952 
9996 

352  COMMON    LOGARITHMS. 

To  Find  the  Antilogarithm  of  a  Given  Logarithm. 

647.  If  the  given  mantissa  can  be  found  in  the  table, 
the  first  two  figures  of  the  required  number  are  in  the 
column  headed  N  on  the  same  line  with  the  mantissa,  and 
the  third  figure  is  at  the  top  of  the  column  that  contains 
the  mantissa. 

The  position  of  the  decimal  point  is  determined  by  the 
characteristic  (§  646). 

Thus,  the  number  corresponding  to  the  logarithm 
2.9736  is  941,  0.8169  is  6.56, 

6.0899  is  1,230,000,  7.8739  -  10  is  0.00748. 

648.  If  the  given  mantissa  cannot  be  found  in  the  table, 
find  in  the  table  the  two  adjacent  mantissas  between  which 
the  given  mantissa  lies,  and  the  three  figures  corresponding 
to  the  smaller  of  these  two  mantissas  are  the  first  three 
significant  figures  of  the  required  number.  The  fourth 
figure  is  found  as  in  the  following 

Examples.  1.  Find  the  antilogarithm  of  the  logarithm 
3.7936. 

Solution.  The  two  adjacent  mantissas  of  the  table  between  which 
the  given  mantissa  7936  lies  are  7931  and  7938.  The  corresponding 
numbers  are  621  and  622.  The  smaller  of  these,  621,  contains  the 
first  three  significant  figures  of  the  required  number. 

The  difference  between  the  two  adjacent  mantissas  is  7,  and  the 
difference  between  the  corresponding  numbers  is  1. 

The  difference  between  the  smaller  of  the  two  adjacent  mantissas, 
7931,  and  the  given  mantissa,  7936,  is  6.  Therefore,  the  number  to  be 
annexed  to  621  is  \  of  1  =  0.71,  and  the  fourth  significant  figure  of 
the  required  number  is  7. 

Hence,  the  required  number  is  6217. 

2.   Find  the  antilogarithm  of  the  logarithm  7.3884  — 10. 

Solution.  The  two  adjacent  mantissas  of  the  table  between  which 
the  given  mantissa  3884  lies  are  3874  and  3892.  The  corresponding 
numbers  are  244  and  246.  The  smaller  of  these,  244,  contains  the 
first  three  significant  figures  of  the  required  number. 


COMMON    LOGARITHMS.  353 

The  difference  between  the  two  adjacent  mantissas  is  18,  and  the 
difference  between  the  corresponding  numbers  is  1. 

The  difference  between  the  smaller  of  the  two  adjacent  mantissas, 
3874,  and  the  given  mantissa,  3884,  is  10.  Therefore,  the  number  to 
be  annexed  to  244  is  f  §  of  1  =  0.55,  and  the  fourth  significant  figure 
of  the  required  number  is  6. 

Hence,  the  required  number  is  0.002446. 

Exercise  152. 
Find  the  antilogarithms  of  the  following  logarithms  : 


1. 

3.9017. 

7. 

2.9850. 

13. 

8.7324-10. 

2. 

1.2076. 

8. 

4.5388. 

14. 

9.5555  - 10. 

3. 

0.4442. 

9. 

0.8550. 

15. 

6.0216  - 10. 

4. 

1.0090. 

10. 

9.9992  - 10. 

16. 

7.0080  - 10. 

5. 

4.8697. 

11. 

7.0016  - 10. 

17. 

8.2361  - 10. 

6. 

1.9214. 

12. 

9.2618  - 10. 

18. 

9.4513  - 10. 

649.  Since  every  factor  of  a  product  may  be  expressed 
as  a  power  of  ten  (§  628), 

The  logarithm  of  a  product  is  equal  to  the  sum  of  the  loga- 
rithms of  its  factors  (§  69). 

650.  Example.      Find  by  logarithms   the  product  of 
908.4  X  0.05392  X  2.117. 

Solution.  log  908.4      =  2.9583 

log  0.05392  =  8.7318  -  10 

log  2.117      =  0.3257 

2.0158  =  log  103.7. 
Therefore,  the  required  product  is  103.7. 

Exercise  153. 
Find  by  logarithms  the  value  of  : 

1.  948.22  X  0.4387.  4.    270.05  X  0.0087. 

2.  1.9704  X  0.0786.      5.  11.163  X  0.3333. 
8.  380.25  X  0.00673.     6.  777.78  X  0.0787. 


354  COMMON    LOGARITHMS. 

7.  216.21  X  0.76312.  11.  2.6537  X  0.2313. 

8.  0.56127X1.2312.  12.  37.587X12.371. 

9.  0.86311X56.371.  13.  89.313X2.3781. 
10.  59.795  X  0.7955.  14.  9.1765  X  0.089. 

15.  4786  X  5.4187  X  0.00218  X  0.8652. 

16.  3.1416  X  7.77  X  184  X  0.01865. 

17.  0.7854X129.6X63.45X0.0021. 

18.  1842.65X9.876X0.843X0.0265. 

19.  12.48  X  44.63  X  32.78  X  0.004587. 

20.  0.9876  X  0.8765  X  0.7654  X  0.6543. 

651.  Any  required  power  of  a  given  power  of  a  number 

is  found  by  multiplying  the  exponent  of  the  given  power 

by  the  exponent  of  the  required  power. 

Thus,  the  cube  of  102  =  102><3  =  106;  the  fifth  power  of  10*  = 
104x5  =  loao.     Hence, 

The  logarithm  of  a  power  of  a  number  is  found  by  multi- 
plying the  logarithm  of  the  number  by  the  exponent  of  the 
power. 

652.  Example.     Find  by  logarithms  the  cube  of  0.0497. 

Solution.  log  0.0497   =  8.6964  —  10 

8 
log  0.04973  =  6.0892  -  10  =  log  0.0001228. 
Therefore,  0.04978  =  0.0001228. 

Notk.  The  real  product  is  26.0892  —  30.  Removing  20  from  the 
26  and  —  20  from  the  —  30,  we  have  6.0892  —  10. 

Exercise  154. 
Find  by  logarithms  the  value  of  : 

1.  5.06s.  6.    0.7685*.  11.  2.8614154. 

2.  2.5016.  7.    0.96118.  12.  3.791256. 

3.  1.7167.  8.    0.02312.  13.  0.021875*. 

4.  1.17810.  9.   0.85673.  14.  0.871527. 

5.  7.68216.  10.    0.54386.  15.  0.959568. 


COMMON    LOGARITHMS.  355 

653.  Any  required  root  of  a  given  power  of  a  number 
is  found  by  dividing  the  exponent  of  the  power  by  the 
index  of  the  root. 

Thus,  the  sixth  root  of  107,  that  is,  VlO7  =  10*.     Hence, 

The  logarithm  of  a  root  of  a  number  is  found  by  dividing 
the  logarithm  of  the  number  by  the  index  of  the  root. 

654.  Examples.     1.    Find  the  cube  root  of  271. 

Solution.  log  271  =    2.4330 

Divide  by  3,  3)2.4330 

0.8110  =  log  6.471. 

Therefore,  ^271  =  6.471. 

2.    Find  the  fifth  root  of  0.4654. 

Solution.  log  0.4654  =    9.6679-10 

Add,  40.  -  40 

Divide  by  5,  5)49.6679-50 

9.9336 -10  =  log  0.8582. 

Therefore,  ^0.4654  =  0.8582. 

If  the  given  number  is  less  than  1,  its  logarithm,  when  written  in 
the  ordinary  form,  will  have  a  —  10  annexed.  In  this  case,  the  form 
of  the  logarithm  should  be  changed  as  in  Example  2,  so  that,  when 
the  logarithm  is  divided  by  the  index  of  the  root,  the  negative  number 
of  the  quotient  shall  be  —  10. 

Exercise  155. 
Find  by  logarithms  the  value  of  : 

1.  13*.                       8.   879™.  15.  93.73*. 

2.  29*.                        9.    0.609*.  16.  21.97g. 

3.  471*.                   10.    0.8716*.  17.  7.9351. 

4.  288*.                   11.    0.021641*.  18.  0.815f. 

5.  1019*.                 12.    0.9825*.  19.  2.81451 

6.  1281*.                  13.    0.42184*.  20.  0.04165A. 

7.  1862*.                 14.    0.02187*.  21.  4,516;298T\ 


356  COMMON   LOGARITHMS. 

655.  Since  a  quotient  is  equal  to  the  divideud  divided 
by  the  divisor, 

The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of 
the  dividend  minus  the  logarithm  of  the  divisor  (§  108). 

656.  Example.     Divide  905.6  by  38.45. 

Solution.  log  905.6  =  2.9569 

log  38.45  =  1.5849 

1.3720  =  log  23.55. 
Therefore,  905.6  -r  38.45  =  23.55. 

657.  The  Cologarithm  of  a  Number.  The  logarithm 
of  the  reciprocal  of  a  number  is  called  the  cologarithm  of 
the  number. 

Thus,  colog  30  =  log  3V  =  log  1  —  log  30. 

Since  log  1=0,  log  1  —  log  30  =  —  log  30. 

Now,  in  the  expression  —  log  30,  the  minus  sign  affects 
the  entire  logarithm,  both  the  characteristic  and  the  man- 
tissa. To  avoid  a  negative  mantissa,  we  can  substitute 
for  —  log  30  its  equivalent 

(10  -  log  30)  -  10. 

Therefore,  the  cologarithm  of  a  number  is  found  by  sub- 
tracting the  logarithm  of  the  number  from  10,  and  annex- 
ing —  10  to  the  remainder. 

Note.  The  best  way  to  perform  the  subtraction  is  to  begin  at  the 
left,  subtract  from  9  each  figure  except  the  last  significant  figure,  and 
subtract  the  last  significant  figure  from  10. 

658.  Examples.     1.   Find  the  cologarithm  of  4007. 

Solution.  10.         — 10 

log  4007  =    3.6028 
colog  4007=    6.3972-10 

2.    Find  the  cologarithm  of  0.004007. 
Solution.  10.         — 10 

log  0.004007  =    7.6028  -  10 
colog  0.004007  =   2.3972 


COMMON   LOGARITHMS.  357 

659.  By  using  cologarithms,  the  inconvenience  of  sub- 
tracting the  logarithm  of  the  divisor  is  avoided. 

Thus,  the  example  of  §  656  is  usually  solved  as  follows : 

log  905.6  =  2.9569 
colog  38.45  =  8.4151  -  10 

1.3720  as  log  23.55. 

Therefore,  905.6  -f  38.45  =  23.55. 

660.  Example.     By  the  use  of  cologarithms  find  the 

7.56  X  4667  X  567 
value  ol  899 ^  x  0  00337  x  23435' 

Solution.  log  7.56       =  0.8785 

log  4667       =  3.6690 

log  567         =  2.7536 

colog  899.1      =7.0462-10 

colog  0.00337  =  2.4724 

colog  23,435    =  5.6301  -  10 

2.4498  =  log  281.7. 

Note.  Log  899.1  is  2.9538,  and  its  cologarithm  is  (10-2.9538)— 10, 
or  7.0462  —  10  ;  log  0.00337  is  7.5276  —  10,  and  subtracting  this  from 
10  —  10  we  obtain  for  its  cologarithm  2.4724 ;  log  23,435  is  4.3699,  and 
its  cologarithm  is  (10  —  4.3699)  —  10,  or  5.6301  —  10. 

Exercise  156. 
Find  by  logarithms  the  value  of : 

56.407  _     75.46  X  0.0765 


1. 


13.045  '    93.08  X  98.071 


857.06  98  X  537  X  0.0079 

*  3079.8*  '      67309  X  0.0947 
0.9387  314  X  7.18  X  8132 

'     598.6  '  '    519X827X3.215' 

3069  212  X  2.16  X  8002 

*  0.7891*  '   536X351  X  7.256* 

9.    (fj)«         11.  (5A)2-         13.    (fi-f)5.  15.    (|||)3. 

10.    (if)«.         12.  (4^)*.         14.    (£3)8.  16.    (HH) 


358  COMMON    LOGARITHMS. 

19.258  X  3.1416  X  812.72 
716.4  X  8.002  X  21.465 
2018  X  0.00261  X  1728 

1412  X  0.0965X0.08621* 
44,816  X  17.265  X  181 

28,754  X  1.2871  X  206.45* 
216.1  X  5280  X  144.2 

187.42  X  4622.6  X  156.8* 

5982.55  X  0.02987  X  0.9852 

42.875  X  34.62  X  28.47 
14.718  X  48.67  X  96.542 

2746.2  X  0.0467  X  2.1876* 

/  83.25  X  4267  X  0.008576" 
'    ^0.0327  X  687.5  X  0.005003* 

*/4.1632Xl7.744X  0.7183* 
^3.0132  X  34.34  X  0.08137*' 

7132  X  9.245  X  0.5477s 


17. 
18. 
19. 
20. 
21. 
22. 
23 


25  V™ 


93  X  0.000173*  X  0.01 

5  /65.022  X  0.002753  X  97.98* 
'    "V  7.298  X  0.04754  X  8.1562 ' 

«/23.792  X  0.00756  X  0.4648* 
'    "V  4723*  X  0.6571  X  0.8246* 

6012  X  0.6012*  X  0.6012* 
5926X0.5926*  X  0.5926* 


/0.03214  X  3.7183  X  0.07824*\f 
'    \  0.05142  X  0.4728*  X  1.239s  / 

/  0.07986  X  0.7555*  X  0.5557*  \f 
'    \0.06897  X  0.5777*  X  0.05698V 

/  0.07543  X  0.7689*  X  0.89652  \* 
"    V  0.06987  X  0.07986*  X  0.9867V 


CHAPTEK   XX. 

APPLICATIONS  OF  LOGARITHMS. 
Compound  Interest  Problems. 

661.  In  compound  interest,  if  P  represents  the  principal, 
r  the  rate  per  cent,  n  the  number  of  years,  A  the  amount, 
then       (1  -4-  r)    represents  the  amount  of  $1  for  1  year. 

(1  -f-  r)2  represents  the  amount  of  $1   for  2  years. 

(1  +  r)8  represents  the  amount  of  $1   for  3  years. 

(1  -+■  r)n  represents  the  amount  of  $1   for  n  years. 

P  X  (1  +  r)n  represents  the  amount  of  $P  for  n  years. 

Therefore,  A  =  P  X(l  +  r)n, 

and  log  A  =  log  P-\-n  X  log  (1  +  r). 

662.  Examples.     1.    Find  the  amount  of  $150  for  6 
years  at  4%  compound  interest. 

Solution.     Here  P  =  160,  r  =  0.04,  n  =  6. 
Therefore,        log  .A  =  log  150  +  6  X  log  1.04. 
log  150  =  2.1761 
6  X  log  1.04  =  0.1020 

log  A  -  2.2781  =  log  189. 7. 
Hence,  the  required  amount  is  $189.70. 

2.    What  principal  will  amount  to  $500  in  5  years  at 
^\°f0  compound  interest  ? 

Solution.     Here  A  =  500,  r  =  0.045,  n  =  5. 
Therefore,     500  =  P  X  (1.045)5 ;  and  P  =  -^^  • 

log  500     =  2.6990 
5  X  colog  1.045  =  9.9045  -  10 

log  P  =  2.6035  =  log  401.4. 

Hence,  the  required  principal  is  $401.40. 


360  COMPOUND   INTEREST    PROBLEMS. 

3.  At  what  rate  of  interest  will  $360  amount  to  $481.80 
in  5  years  at  compound  interest  ? 

Solution.     Here  A  -  4S1.80,  P  =  300,  n  =  5. 

Therefore,  481.80  =  360  X  (1  +  r)6, 

„    ,     ,r      481.80         _    4   .  s/481.80 

log  481.80=    2.6828 
colog360       =    7.4437  -  10 
5)0.1265 
log  (1  +  r)  =    0.0253  =  log  1.06. 

Hence,  the  required  rate  of  interest  is  6%. 

4.  In  what  time  at  3£%  compound  interest  will  $540 
amount  to  $619.40  ? 

Solution.     Here  A  =  619.40,  P  =  540,  r  =  0.035. 
Therefore,  log  619.40  =  log  540  +  n  X  log  1.035, 

n  X  log  1.035  =  log  619.40  +  colog  540, 

log  619.40  4-  colog  540 

and  n  =       \ ,  MC     " 

log  1.035 

_  2.7920  +  7.2676  -10 

0.0149 

0.0596 


0.0149 
Hence,  the  required  time  is  4  years. 


=  4. 


Exercise  157. 

1.  Find  the  compound  interest  on  $1280  for  7  years 
at  4£%. 

2.  Find  the  compound  interest  on  $2645  for  5  years 
at  3£%. 

3.  Find  the  amount  of  $848  for  6  years  at  5%  com- 
pound interest. 

4.  Find  the  amount  of  $3600  for  5  years  at  5£%  com" 
pound  interest. 

5.  What  principal  will  amount  to  $720  in  4  years  at 
6%  compound  interest  ? 


COMPOUND    INTEREST    PROBLEMS.  361 

6.  What  principal  will  amount  to  $1640  in  6  years  at 
3%  compound  interest? 

7.  At  what  rate  of  interest  will  $648  amount  to  $788.20 
in  5  years  at  compound  interest  ? 

8.  At  what  rate  of  interest  will  $2415  amount  to  $3237 
in  6  years  at  compound  interest  ? 

9.  In  what  time  at  4-j-%  compound  interest  will  $1265 
amount  to  $1576  ? 

10.  In  what  time  at  5%  compound  interest  will  $1845 
amount  to  $2413  ? 

663.  The  table  on  the  following  page  has  been  made 
with  the  aid  of  logarithms.  It  shows  the  amount  of  $1  at 
compound  interest  at  various  per  cents  for  from  1  to  20 
years.  The  compound  interest  on  $1  is  found  by  subtract- 
ing 1  from  the  proper  number  shown  in  the  table. 

664.  Examples.  1.  What  principal  will  in  10  years 
at  6%  compound  interest  yield  $1898.04  interest  ? 

Solution.     The  interest  on  $1  for  10  yr.  at  6%  is  $0.79085. 
Since  $0.79085  is  the  interest  on  $1, 

$1898.04  is  the  interest  on  $^^|,  or  $2400. 
0.79085 

2.  In  what  time  will  $1600  at  k\°j0  compound  interest 
yield  $1000  interest  ? 

Solution.  Since  $1600  yields  $1000,  $1  will  yield  T^  of  $1000, 
or  $0,625,  in  the  same  time,  and  $1  will  amount  to  $1,625.  By  the 
table,  $1  will  in  11  yr.  at  4£%  amount  to  $1.62285,  and  in  12  yr.  to 
$1.69588.     Hence,  the  required  time  is  a  little  more  than  11  yr. 

3.  At  what  rate,  compound  interest,  will  $1500  yield 
$1201.41  interest  in  15  years  ? 

Solution.  Since  $1500  yields  $1201.41  interest  in  15  yr.,  $1  in 
15  yr.  will  yield  T^  of  $1201.41,  or  $0.80094,  and  $1  will  amount 
in  15  yr.  to  $1.80094.  In  the  table,  opposite  15  yr.,  we  find  in  the  4% 
column  the  amount  of  $1  is  $1.80094.     Therefore,  the  rate  is  4%. 


362 


COMPOUND    INTEREST    PROBLEMS. 


665.  Table. 

Showing  the  amount  of  $1  at  compound  interest  for: 


Vlt. 

2  PER  CENT. 

2£  PER  CENT. 

3  PER  CENT. 

31  PER  CENT. 

4  PER  CENT. 

1 

1.02000 

1.02500 

1.03000 

1.03500 

1.04000 

2 

1.04040 

1.05063 

1.06090 

1.07123 

1.08160 

3 

1.06121 

1.07689 

1.09273 

1.10872 

1.12486 

4 

1.08243 

1.10381 

1.12551 

1.14752 

1.16986 

5 

1.10408 

1.1.5141 

1.15927 

1.18769 

1.21665 

6 

1.12616 

1.15969 

1.19405 

1.22926 

1.26532 

7 

1.14869 

1.18869 

1.22987 

1.27228 

1.81608 

,  8 

1.17166 

1.21840 

1.2(5677 

1.31681 

1.36857 

9 

1.19509 

1.24886 

1.30477 

1.36290 

1.42:;::  1 

10 

1.21899 

1.28009 

1.34392 

1.41060 

1.48024 

11 

1.24337 

1.31209 

1.381J:5 

1.45997 

1.53945 

12 

1.26824 

1.34489 

1.42576 

1.51107 

1.60103 

13 

1.29961 

1.37851 

1.46853 

1.56396 

1.(5(5507 

14 

1.31948 

1.41297 

1.51259 

1.61870 

1.7:5168 

15 

1.34587 

1.44830 

1.65797 

1.67535 

1.80094 

16 

1.37279 

1.48451 

1.60471 

1.73399 

1.87208 

17 

1.40024 

1.52162 

1.66286 

1.794(58 

1.04790 

18 

1.42825 

1.55966 

1.70248 

1.85749 

2.0268S 

19 

1.45681 

1.59865 

1.75351 

1.92250 

2.10685 

20 

1.48595 

1.63862 

1.80611 

1.98979 

2.19112 

YB. 

4J  PER  CENT. 

5  PER  CENT. 

5J  PER  CENT. 

6  PER  CENT. 

7  PER  CENT. 

1 

1.04500 

1.05000 

1.05500 

1.06000 

1.07000 

2 

1.09203 

1.10260 

1.11303 

1.12360 

1.1 441  to 

3 

1.14117 

1.16763 

1.17424 

1.19102 

1.22504 

4 

1.19252 

1.21551 

1.23882 

1.26248 

1.31080 

5 

1.24618 

1.27628 

1.80606 

1.33823 

1.40255 

6 

1.80296 

1.34010 

1.37884 

1.41852 

1.50073 

7 

1.36086 

1.40710 

1.46468 

1.60363 

1.60578 

8 

1.42210 

1.47746 

1.53469 

1.69385 

1.71819 

9 

1.48610 

1.55133 

1.61909 

1.(58948 

1.83846 

10 

1.55297 

1.62889 

1.70814 

1.79085 

1.96715 

11 

1.62285 

1.71034 

1.80209 

1.89830 

2.10485 

12 

1.69588 

1.79586 

1.90121 

2.01220 

2.25210 

13 

1.77220 

1.88565 

2.00577 

2.13293 

2.40986 

14 

1.851H4 

1.97993 

2.11609 

2.2(5090 

2.57853 

15 

1.93528 

2.07893 

2.23248 

2.80666 

2.75903 

16 

2.02237 

2.18287 

2.35626 

2.54035 

2.95216 

17 

2.11338 

2.29202 

2.48480 

2.69277 

3.15882 

18 

2.20848 

2.40662 

2.62147 

2.85434 

3.37993 

19 

2.30786 

2.52695 

2.76565 

3.02660 

3.61(55:5 

20 

2.41171 

2.65330 

2.91776 

3.20714 

3.86968 

COMPOUND    INTEREST    PROBLEMS.  363 


Exercise  158. 

1.  A  man  deposits  $60  in  a  savings  bank,  and  draws 
out  his  whole  account  at  the  end.  of  8  years,  with  4% 
compound  interest.     What  amount  does  he  receive  ? 

2.  What  will  $100  amount  to  in  7  years  with  interest 
at  8%  per  annum,  compounded  semi-annually  ? 

3.  In  how  many  years  will  a  sum  of  money  double 
itself  at  6%,  compounded  annually  ? 

4.  In  how  many  years  will  a  sum  of  money  treble 
itself  at  6%,  compounded  annually  ? 

5.  In  how  many  years  will  $87  amount  to  $99  at  3%, 
compounded  annually  ? 

6.  In  how  many  years  will  $100  amount  to  $175  at 
4%,  compounded  annually  ? 

7.  At  what  rate  per  cent  will  a  sum  of  money  double 
itself  in  12  years,  compound  interest  ? 

8.  At  what  rate  will  a  sum  of  money  treble  itself  in 
19  years,  compound  interest  ? 

9.  At  what  rate  will  $80  at  compound  interest  amount 
to  $110  in  8  years  ? 

10.  What  sum  must  be  invested  at  5%,  compound  in- 
terest, to  amount  to  $1200  in  7  years  ? 

11., What  sum  must  be  invested  at  4%,  compound  in- 
terest, to  amount  to  $2000  in  10  years?.  To  amount  to 
$5000  in  8  years  ? 

12.  At  what  rate  compound  interest  will  $462.50  yield 
$277.98  interest  in  12  years  ? 

13.  What  principal  will  in  10  years  at  6%  amount  to 
$3612.22,  interest  being  compounded  semi-annually  ? 

14.  In  what  time  at  5%  will  $1250  amount  to  $2000, 
interest  being  compounded  semi-annually  ? 

1 5.  At  what  rate  per  annum  will  $500  amount  to  $779.83 
in  9  years,  interest  being  compounded  semi-annually  ? 


364  ANNUITIES. 


Annuities. 


666.  An  annuity  is  a  sum  of  money  to  be  paid  at  reg- 
ular intervals  of  time,  as  years,  half  years,  quarter  years. 

667.  A  perpetual  annuity  is  an  annuity  that  continues 
forever. 

668.  A  certain  annuity  is  an  annuity  that  begins  at  a 
specified  time  and  ends  at  a  specified  time. 

669.  A  contingent  annuity  is  an  annuity  that  depends 
upon  some  particular  event,  as  the  death  of  an  individual. 

Life  insurance,  dowers,  and  pensions  are  examples. 

670.  The  final  value  of  an  annuity  is  the  sum  to  which 
all  its  payments  at  compound  interest  will  amount  at  the 
end  of  the  annuity. 

671.  The  present  value  of  an  annuity  is  the  sum  which 
at  compound  interest  will  amount  to  its  final  value. 

672.  A  sinking  fund  is  the  final  value  of  sums  of  money 
set  apart  at  regular  intervals  of  time,  and  invested  at  com- 
pound interest,  to  pay  a  debt  due  at  a  stated  time. 

673.  To  Find  the  Final  Value  of  an  Annuity. 

If  S  represents  the  amount  paid  each  year,  72  the  amount 
of  $  1  at  interest  for  1  year,  n  the  number  of  years,  and  A 
the  final  value,  then, 

The  amount  at  the  end  of  the 
1st  year  =  S, 
2d    year  =  S+SR, 
3d    yea,r=  S+SR+SR2, 

nth  year  =  S  +  SR  +  SR2  + +  /SB—1. 

That  is,  A  =  S+SR+SR2  + +  SE*-\        (1) 

Multiplying  (1)  by  R,  we  have 

AR  =  SR  +  SR2  +  SMS  + +  SB—1  +  S&.       (2) 


ANNUITIES.  365 

Subtracting  (1)  from  (2),  we  have 

AR-A  =  SRn-  S, 
or  A(R-l)  =  S(Rn-l). 

Therefore,  A  =  — ^ — *  • 

K  —  1 

Here,  R  —  1  is  the  interest  on  $1  for  1  year,  and  Rn  —  1  is  the 
compound  interest  of  $1  for  n  years. 

674.   To  Find  the  Present  Worth  of  an  Annuity. 

If  P  represents  the  present  worth  of  the  annuity,  then 
the  amount  of  P  for  n  years  =  A,  the  final  value  of  the 
annuity  in  n  years. 

The  amount  of  P  for  n  years 

=  P  (1  +  r)»  =  PR%  (§  661) 

and  -*-*£^-  (§673) 

Therefore,  PR* 

JX  —  J- 

Rn-1 


R  — 

1 

S(Rn- 

"^ 

R- 

1       ' 

S(Rn- 

"1)_ 

8 

Rn(R- 

p.  wri  t.t.p.n 

-1) 

s 

R- 

1 

1  - 

and  P  =  — ^— — -f  =  - X 


i2" 


The  value  of  P  may  be  written  — X  (  1  —  —  )  • 

■  R  —  l\        RnJ 

If  the  annuity  is  perpetual,  n  is  too  large  to  be  counted  ; 
therefore,  Rn  is  too  large  to  be  counted  and  —  =  0. 
Hence,  P==_^  =  |. 

675.    Examples.     1.    What  is  the  value  of  a  sinking 

fund,  if  $6000  is  set  apart  annually  for  6  years  and  put  at 

4%  compound  interest  ? 

.       S(R»-1)      $6000  x  (1.046-1) 
Solution.        A  =     VR_1  ;-  =  - ^1 * •  (§  673) 

By  the  use  of  logarithms,  A  is  found  to  be  $39,750. 


366  ANNUITIES. 

2.  Find  the  present  value  of  an  annuity  of  $500  for 

5  years,  if  money  is  worth  4%. 

8  R»-l       500       1.045-1         tmm-M 

SoumoH.         P=_x-^=  —  X-j^p-       (§674) 

log  1.04  =0.0170 
5 

log  1.045  =  0.0850  =  log  1.216. 
™      r  n      500       0.216 

Therefore,  P  =  0^4*  1^16 ' 

log  500     =  2.6990 
log  0.216  =  9.3345  -  10 
colog  0.04    =  1.3979 
colog  1.216  =  9.9150-10 

log  P=  3.3464  =  log  2220. 

Hence,  the  present  value  of  the  annuity  is  §2220. 

3.  Find  the  present  value  of  a  perpetual  annuity  of 
$500,  if  money  is  worth  4%. 

Solution.  P  =  f=  O^ol  =  $12>500-  <§  674> 

Exercise  159. 

1.  Find  the  present  value  of  an  annuity  of  $300  for 

6  years,  if  money  is  worth  5%. 

2.  Find  the  present  value  of  an  annuity  of  $600  for 

4  years,  if  money  is  worth  5£%. 

3.  Find  the  present  value  of  an  annuity  of  $800  for 

5  years,  if  money  is  worth  6%. 

4.  Find  the  present  value  of  a  perpetual  scholarship  of 
$900,  if  money  is  worth  3£%. 

5.  Find  the  present  value  of  a  perpetual  fellowship  of 
$3200,  if  money  is  worth  4£%. 

6.  What  is  the  value  of  a  sinking  fund,  if  $25,000  is 
set  apart  yearly  for  7  years  at  ±\°f0  compound  interest  ? 

7.  What  is  the  value  of  a  sinking  fund,  if  $18,000  is 
set  apart  yearly  for  5  years  at  3£%  compound  interest? 


an;n  UiTIES. 


367 


676.  The  table  at  the  bottom  of  the  page  shows  the 
average  number  of  years  persons  live  after  the  ages  indi- 
cated. This  table  is  known  as  the  Carlisle  Table  because 
it  is  based  upon  the  rate  of  mortality,  as  carefully  observed 
at  Carlisle,  England.  Several  other  tables  of  Expectancy 
of  Life  have  been  compiled  from  other  data  and  are  in 
common  use  with  insurance  companies. 

The  table  on  the  following  page  has  been  made  with 
the  aid  of  logarithms.  It  shows  the  present  value  of  an 
annuity  of  $1  per  annum  at  compound  interest  from  1  to 
40  years  at  3£%  and  at  4%. 


677. 


Carlisle  Table  of  Expectancy  of  Life. 


EX- 

EX- 

EX- 

EX- 

AGE. 

PECTANCY. 

AGE. 

PECTANCY. 

AGE. 

PECTANCY 

AGE. 

PECTANCY. 

0 

38.72 

26 

37.14 

52 

19.68 

78 

6.12 

1 

44.68 

27 

36.41 

53 

18.97 

79 

5.80 

2 

47.55 

28 

35.69 

54 

18.28 

80 

5.51 

3 

49.82 

29 

35.00 

55 

17.58 

81 

5.21 

4 

50.76 

30 

34.34 

56 

16.89 

82 

4.93 

6 

51.25 

31 

33.68 

57 

16.21 

83 

4.65 

6 

51.17 

32 

33.03 

58 

15.55 

84 

4.39 

7 

50.80 

33 

32.36 

59 

14.92 

85 

4.12 

8 

50.24 

34 

31.68 

60 

14.34 

86 

3.90 

9 

49.57 

35  . 

31.00 

61 

13.82 

87 

3.71 

10 

48.82 

36 

30.32 

62 

13.31 

88 

3.59 

11 

-48.04 

37 

29.64 

63 

12.81 

89 

3.47 

12 

47.27 

38 

28.96 

64 

12.30 

90 

3.28 

13 

46.51 

39 

28.28 

65 

11.79 

91 

3.26 

14 

45.75 

40 

27.61 

66 

11.27 

92 

3.37 

15 

45.00 

41 

26.97 

67 

10.75 

93 

3.48 

16 

44.27 

42 

26.34 

68 

10.23 

94 

3.53 

17 

43.57 

43 

25.71 

69 

9.70 

95 

3.53 

18 

42.87 

44 

25.09 

70 

9.18 

3.46 

19 

42.17 

45 

24.46 

71 

8.65 

97 

3.28 

20 

41.46 

46 

23.82 

72 

8.16 

98 

3.07 

21 

40.75 

47 

23.17 

73 

7.72 

99 

2.77 

22 

40.04 

48 

22.50 

74 

7.33 

100 

2.28 

23 

39.31 

49 

21.81 

75 

7.01 

101 

1.79 

24 

38.59 

50 

21.11 

76 

6.69 

102 

1.30 

25 

37.86 

51 

20.39 

77 

6.40 

103 

0.83 

368 


ANNUITIES. 


678. 


Table. 


Showing  the  present  value  of  an  annuity  of  $1  per  annum, 
at  compound  interest  from  1  to  Jfi  years  at  3^°f0  and  at  4°/o- 


YR. 

3J  PER  CENT. 

4  PER  CENT. 

YR. 

3£  PER  CENT. 

4  PER  CENT. 

1 

0.96018 

0.90154 

21 

14.09797 

14.02910 

2 

1.89909 

1.88010 

22 

15.10713 

14  45112 

3 

2.80164 

2.77609 

23 

15.02041 

14.85084 

4 

3.03708 

8.62980 

24 

16.06887 

16.24606 

6 

4.51505 

4.45182 

■25 

10.48152 

15.02208 

6 

6.32855 

5.24214 

26 

10.89035 

15.98277 

7 

0.11454 

c.  00206 

27 

17.28537 

16.82060 

8 

0.87390 

0.73275 

28 

17.00702 

10.00300 

9 

7.00709 

7.43533 

29 

18.03577 

10.98372 

10 

8.31661 

8.11090 

80 

18.39205 

17.29203 

11 

9.00155 

8.7004H 

81 

18.73028 

17.58849 

12 

9.00333 

8.88607 

32 

19.00887 

17.87355 

13 

10.30274 

0.06666 

88 

19.39021 

18.14766 

14 

10.92052 

10.50312 

34 

19.70008 

18.41120 

15 

11.51741 

11.11839 

35 

20.00000 

18.00401 

16 

12.09412 

11.05230 

30 

20.29049 

18.90828 

17 

12.66188 

12.10507 

37 

20.57053 

19.14258 

18 

13.18908 

12.05930 

38 

20.84109 

19.30780 

19 

13.70984 

13.13394 

39 

21.10250 

19.58449 

20 

14.21240 

13.69033 

40 

21.35507 

19.79277 

679.  Examples.  1.  Find  the  present  value  of  an 
annuity  for  $500  for  5  yr.  at  4%. 

Solution.  The  present  value  of  81  for  5  yr.  at  4%  by  the  table  is 
$4.45182  ;   and  of  $500  is  500  X  $4.45182,  or  $2225.01. 

2.  A  person  41  years  of  age  pays  $9797.75  for  a  life 
annuity.  If  interest  is  reckoned  at  4%,  find  the  amount  of 
the  annuity. 

Solution.     According  to  the  table  on  page  367,  the  expectancy  of 

life  for  a  person  41  years  of  age  is  about  27  years. 

The  present  value  of  an  annuity  of  $1  for  27  yr.  at  4%  is  $16.32959. 

9797  75 
Hence,  the  amount  of  the  annuity  is  $  '      t  or  $600. 

10. 32959 


ANNUITIES.  369 


Exercise  160. 

1.  Find  the  present  value  of  an  annuity  of  $900  for 
15  years  at  4%. 

2.  Find  the  present  value  of  an  annuity  of  $1500  for 
12  years  at  4%. 

3.  Find  the  present  value  of  an  annual  pension  of  $144 
for  10  years  at  3J%. 

4.  Find  the  present  value  of  a  scholarship  of  $200  for 
25  years  at  2>\°f0. 

5.  Find  the  present  value  of  an  annuity  of  $2500  for 
30  years  at  4%. 

6.  Find  the  present  value  of  an  annuity  of  $250  for 
12  years  at  3£%. 

7.  A  person  22  years  old  has  a  life  annuity  of  $750. 
Find  its  present  value  at  4%. 

8.  A  person  35  years  old  has  a  life  annuity  of  $1800. 
Find  its  present  value  at  4%. 

9.  A  person  53  years  old  has  a  life  annuity  of  $500. 
Find  its  present  value  at  4%. 

10.  A  person  75  years  old  has  a  life  annuity  of  $2400. 
Find  its  present  value  at  3£%. 

11.  A  boy  15  years  old  has  a  life  annuity  of  $3250. 
Find  jts  present  value  at  4%. 

12.  A  person  22  years  old  pays  $4948.19  for  a  life 
annuity.     If  interest  is  4%,  find  the  amount  of  the  annuity. 

13.  A  person  29  years  old  pays  $7465.84  for  a  life 
annuity.     If  interest  is  4  °f0 ,  find  the  amount  of  the  annuity. 

14.  A  person  35  years  old  pays  $9368.14  for  a  life 
annuity.  If  interest  is  3£%,  find  the  amount  of  the 
annuity. 

15.  A  person  44  years  old  pays  $5933.35  for  a  life 
annuity.  If  interest  is  3£%,  find  the  amount  of  the 
annuity. 


370  COOPERATIVE   BANKS. 

Cooperative  Banks. 

680.  A  cooperative  bank  is  a  mutual  corporation  with 
the  object  of  the  accumulation  of  a  capital  to  be  loaned  to 
its  members,  especially  for  the  purchase  of  homes. 

681.  Shares.  The  capital  stock  is  usually  divided  into 
shares  of  final  value  $200  each,  that  are  paid  for  in  monthly 
instalments  of  $1  each.  The  number  of  shares  that  any 
member  may  purchase  is  limited,  usually  to  twenty-five. 
Each  shareholder  pays  $1  a  month  per  share  until  his 
shares  are  worth  $200  each.  The  shares  are  then  said  to 
be  matured.  At  maturity  the  shareholder  receives  $200 
in  money  for  each  share  he  holds. 

If  no  profits  were  added  to  the  value  of  the  shares,  it 
would  take  200  months,  that  is,  16  years  8  months  to 
mature  the  shares  ;  but  the  profits  generally  reduce  this 
time  to  between  10  and  12  years. 

682.  Loans.  Any  shareholder  may  borrow  $200  on 
each  share  he  holds,  provided  he  furnishes  the  security 
required  by  law.  Security  may  be  by  mortgage  upon  real 
estate  or  upon  the  shares  themselves.  If  the  shares  are 
offered  as  security,  no  shareholder  is  allowed  to  borrow 
more  than  the  present  value  of  his  shares. 

The  amount  of  a  loan  is  usually  limited  to  $2000. 

683.  When  the  accumulation  of  the  various  payments 
has  reached  a  certain  sum,  these  funds  are  offered  at  auction 
and  loaned  to  the  shareholder  who  offers  proper  security 
and  bids  the  highest  premium,  in  addition  to  interest  at 
the  rate  of  6%  per  annum.  This  interest  and  premium  is 
added  to  the  general  fund,  and  at  stated  times  is  credited 
equally  among  the  various  shares. 

684.  Fines.  To  prevent  payments  falling  into  arrears, 
a  fine,  usually  2  cents  a  month  on  every  dollar  not  paid 


COOPERATIVE   BANKS.  371 

when  due,  is  imposed  on  delinquent  shareholders,  whether 
the  delinquency  is  the  monthly  instalment,  interest,  or 
premium. 

685.  Examples.  1.  Find  the  cost  at  compound  interest 
of  a  cooperative  bank  share,  if  the  share  matured  in  11 
years,  and  money  was  worth  4%. 

Solution.     11  yr.  =  132  mo.     $1  was  paid  monthly  for  132  mo. 
The  rate  of  interest  was  4%  yearly,  or  \°j0  monthly. 

g(g»-l)_*lx(1.003j*»-1) 
A~     R-\      ~  1.003* -1  (§673) 

By  the  four-place  table  of  logarithms,  the  value  of  this  fraction  is 
found  to  be  $163.80  ;  by  a  seven-place  table,  the  value  is  $165.46. 
The  seven-place  table  gives  a  very  close  approximation. 

2.  Find  the  cost  at  compound  interest  of  a  loan  of  $200 
from  a  cooperative  bank,  if  the  borrower  pays  $1  per  month 
interest.  The  shares  are  worth  $60,  and  mature  in  7  years, 
and  money  is  worth  4%. 

Solution.  The  borrower  pays  monthly  $2  for  7  yr.,  that  is,  84  mo. 
The  final  value  of  $2  deposited  monthly  for  84  mo.  at  4%  is  found  by 
§  673  to  be  $191.40,  and  the  compound  amount  of  $60  at  4%  for  7  yr. 
is  $78.96.     Hence,  the  cost  of  the  loan  is  $191.40  +  $78.96  =  $270.36. 

Exercise  161. 

Find  the  cost  at  compound  interest  of  a  : 

1.  Cooperative  bank  share  that  matured  in  10  years, 
when  money  was  worth  ty°/0. 

2.  Cooperative  bank  share  that  matured  in  llj  years, 
when  money  was  worth  6%. 

3.  How  much  more  does  it  cost  to  borrow  $2000  from 
a  cooperative  bank,  monthly  interest  being  $12,  and  the 
shares  maturing  in  10  years,  than  to  borrow  $2000  at  com- 
pound interest  for  10  years,  if  money  is  worth  5°/0  in  both 
cases  ? 


CHAPTER   XXI. 

MISCELLANEOUS  PEOBLEMS. 

In  solving  these  problems,  logarithms  should  be  used  whenever 
they  can  be  used  with  advantage. 

1.  Make  six  different  numbers  with  the  digits  1,  2,  3, 
and  find  their  sum. 

2.  Make  six  different  numbers  with  the  digits  2,  3,  5, 
and  find,  by  logarithms,  their  continued  product. 

3.  Make  six  different  numbers  with  the  digits  8,  7,  3, 
and  find,  by  logarithms,  their  continued  product. 

4.  Find,  by  logarithms,  the  inissing  term  in  each  of  the 
following  proportions  : 

(i)  7.13 : 3.57 : :  4.18 :  ?.        (iii)  7.37 :  ? : :  86.1 :  43.7. 
(ii)  5.89 :  76.3 : :  ?  :  38.7.        (iv)  ? :  69.7 : :  3.79 :  29.4. 

5.  Find,  by  logarithms,  the  value  of  0.08* ;  2734* ; 
21.97*j   ?3-6;   9.71* ;   7.936*. 

_     _.    ,  .,         .         „  J>/4.79*X  3.1416X12.72 

6.  Fmd  the  value  of  V— 0^36^14^8 

7.  If  the  air-line  distance  between  two  points  is  1534  ft., 
and  the  difference  of  level  is  34  ft.,  what  is  the  horizontal 
distance  between  the  two  points  ? 

8.  If  the  road  distance  is  1  mi.,  and  the  rise  347  ft., 
find  the  horizontal  distance. 

9.  If  the  road  distance  is  half  a  mile,  and  the  hori- 
zontal distance  2513  ft.,  find  the  difference  of  level. 

10.  The  diagonal  of  a  rectangular  floor  is  34.6  ft.,  and 
the  width  is  17.8  ft.     Find  the  length  of  the  floor. 

11.  The  height  of  a  tower  on  the  bank  of  a  river  is  55  ft., 
and  the  length  of  a  line  from  the  top  of  the  tower  to  the 
opposite  bank  is  78  ft.     Find  the  breadth  of  the  river. 


MISCELLANEOUS    PKOBLEMS.  873 

12.  The  number  of  seamen  at  Portsmouth  is  800,  at 
Charlestown  404,  and  at  Brooklyn  756.  A  ship  is  com- 
missioned whose  complement  is  490  seamen.  Determine 
the  number  to  be  drafted  from  each  place  to  obtain  a  pro- 
portionate number  from  each. 

13.  Show,  without  division,  that  36,432  contains  8,  9, 
11  as  factors. 

14.  Find  the  smallest  multiplier  that  will  make  47,250 
a  perfect  cube. 

1 5.  Find  the  proper  fraction  that,  when  reduced  to  a  con- 
tinued fraction,  has  for  quotients  1,  3,  5,  7,  2,  4. 

16.  If  the  meter  is  equal  to  1.09362  yd.,  find  a  series  of 
four  fractions  that  will  express  more  and  more  nearly  the 
true  ratio  of  the  meter  to  the  yard. 

17.  Find  the  square  factors  contained  in  33,075. 

18.  The  height  of  St.  Peter's,  Rome,  is  yf^  of  a  mile, 
and  that  of  St.  Paul's,  London,  is  ^y¥  of  a  mile.  How 
many  feet  higher  is  St.  Peter's  than  St.  Paul's  ? 

19.  How  many  days  elapsed  between  the  annular  eclipse 
of  May  15,  1836,  and  that  of  March  15,  1858  ? 

20.  In  a  gale,  a  flagstaff  60  ft.  high  snaps  28.8  ft.  from 
the  bottom  ;  and,  not  being  wholly  broken  off,  the  top 
touches  the  ground.  If  the  ground  is  level,  how  far  is  the 
top  frjnn  the  bottom  ? 

21.  Seventeen  trees  are  standing  in  a  straight  line, 
20  yd.  apart ;  a  man  walks  from  the  first  to  the  second 
and  back,  then  to  the  third  and  back,  and  so  on.  How  far 
does  he  walk  ? 

22.  A  canal  is  14f  mi.  long  and  48  ft.  wide.  At  one 
end  is  a  lock  80  ft.  by  24  ft.,  with  a  fall  of  8  ft.  6  in. 
How  many  barges  can  pass  through  the  lock  before  the 
water  in  the  canal  is  lowered  1  in.? 

23.  Find  the  capacity,  in  liters  and  in  bushels,  of  a  box 
1.7m  long,  87cm  wide,  and  31cm  deep. 


374  MISCELLANEOUS   PROBLEMS. 

24.  Find  the  number  of  kilograms  of  olive  oil,  specific 
gravity  0.915,  required  to  fill  a  rectangular  vessel  2.3m  long, 
1.8m  wide,  and  74cm  deep. 

25.  How  many  tons  in  a  block  of  marble  4  ft.  long, 
34  in.  wide,  17.3  in.  thick,  specific  gravity  2.73  ? 

26.  Find  the  surface  of  a  sphere  18.3  in.  in  diameter. 

27.  Find  the  number  of  acres  in  a  circular  field  213  yd. 
2  ft.  in  diameter. 

28.  How  many  cubic  inches  in  a  10-inch  globe  ?  in  a  20- 
inch  globe  ?     What  is  the  ratio  of  their  volumes  ? 

29.  How  many  balls  3  in.  in  diameter  can  be  cast  from 
a  pig  of  iron  7  ft.  long,  6.7  in.  wide,  3.8  in.  thick,  if  the 
waste  in  melting  and  casting  is  reckoned  at  3J%  ? 

30.  Find  the  difference  in  length,  at  80°  F.,  of  a  glass 
rod  and  a  steel  rod,  each  3  ft.  long  at  0°  C,  if  the  expansion 
at  100°  C.  is  0.00085  for  glass  and  0.0012  for  steel. 

31.  A  grain  of  gold  is  beaten  into  leaf  to  cover  56  sq.  in. 
What  weight  will  be  required  to  gild  the  faces  of  a  cube 
whose  edge  is  3£  ft.? 

32.  What  premium  must  be  paid,  at  the  rate  of  4J%,  for 
insuring  a  vessel  worth  $100,000,  in  order  that  in  the  event 
of  loss  the  owner  may  receive  both  the  value  of  the  ship 
and  the  premium  ? 

33.  By  selling  goods  at  60  cents  a  pound,  8%  is  lost. 
What  advance  must  be  made  in  the  price  to  gain  15%  ? 

34.  The  sharpest  grade  on  Mt.  Washington  liy.  is 
1980  ft.  to  the  mile.  What  fraction  of  a  foot  is  the  rise 
for  each  foot  ?     What  is  the  per  cent  of  grade  ? 

35.  Find  the  square  root,  to  four  decimal  places,  of  the 
reciprocal  of  0.0043. 

36.  The  population  of  a  city  in  1890  was  12,298,  show- 
ing a  decrease  of  8£%  on  its  population  in  1880  ;  in  1880 
there  was  an  increase  of  7£%  on  the  census  of  1870.  What 
was  its  population  in  1870  ? 


MISCELLANEOUS    PROBLEMS.  375 

37.  Find  the  increase  of  income  obtained  by  transferring 
25  shares  of  3%  stock  at  94f  to  4%  stock  at  104f,  broker- 
age \  on  each  transaction. 

38.  Each  person  in  breathing  spoils  the  air  of  a  closed 
room  at  the  rate  of  about  8  cu.  ft.  a  minute.  An  audience 
of  400  persons  enter  a  closed  hall  70  ft.  by  40  ft.,  and  20  ft. 
high.     How  long  will  it  take  them  to  spoil  the  air  ? 

39.  How  long  can  the  windows  and  doors  of  a  school- 
room be  safely  kept  closed  when  occupied  by  50  children, 
if  the  room  is  25  ft.  by  20  ft.  and  10  ft.  high  ? 

40.  A  pays  B  $230  as  the  present  value  of  $300  due  in 
5  years.  Which  gains  by  the  payment,  and  how  much,  if 
interest  is  reckoned  at  5  °f0  compound  interest  ? 

41.  Find  the  quantity  of  coal  required  by  a  steamer  for 
a  voyage  of  4043  mi.,  if  her  rate  per  hour  is  14.04  knots, 
and  her  consumption  of  coal  87  long  tons  per  day. 

42.  Find  the  area  of  a  circular  ring  whose  inner  and 
outer  diameters  are  7.36  in.  and  10.64  in.,  respectively. 

43.  A  and  B  can  do  a  piece  of  work  in  13-J-  days ;  A  and 
C  in  lOf  days  ;  A,  B,  and  C  in  1\  days.  In  how  many 
days  can  A  do  the  work  alone  ? 

44.  If  3  men  working  11  hours  a  day  can  reap  20  A. 
in  11  days,  how  many  men  working  12  hours  a  day  can 
reap-a  field  360  yd.  long  and  320  yd.  broad  in  4  days  ? 

45.  Find  the  area  of  a  triangle  whose  sides  are  12  in., 
5  in.,  and  13  in.,  respectively. 

46.  The  four  sides  of  a  field  measured  in  succession  are 
237  ft.,  253  ft.,  244  ft.,  and  261  ft.,  and  the  diagonal  meas- 
ured from  the  end  of  the  first  side  to  the  end  of  the  third 
side  is  351  ft.     Find  the  area  of  the  field. 

47.  The  four  sides  of  a  field  measured  in  succession  are 
361  ft.,  561  ft.,  443  ft.,  and  357  ft.,  and  the  distance  from 
the  beginning  of  the  first  side  to  the  end  of  the  second  side 
is  682  ft.     Find  the  area  of  the  field, 


376  MISCELLANEOUS    PROBLEMS. 

48.  Find  the  altitude  of  a  triangle,  if  each  side  is  1000  ft. 

49.  Find  the  three  altitudes  of  a  triangle,  if  its  sides  are 
17.8mm,  23.6mm,  and  31.5mm,  respectively. 

50.  How  many  square  inches  in  the  surface  of  a  sphere 
that  has  a  radius  of  12.37  in.? 

5 1 .  Find  the  area  of  the  surface  of  the  largest  globe  that 
can  be  turned  out  from  a  joist  4  in.  by  6  in. 

52.  How  many  cubic  inches  in  a  globe  that  has  a  diam- 
eter of  10  in.? 

53.  If  a  tree  is  round,  and  its  girth  is  17  ft.  6  in.,  find 
its  diameter.  Find  the  area  of  a  cross  section,  and  also  the 
number  of  cubic  feet  in  the  largest  sphere  that  can  be  cut 
from  it. 

54.  Find  the  weight  in  kilograms  and  in  pounds  of  an 
iron  ball  21.5cm  in  diameter,  specific  gravity  7.47  ;  of  a  tin 
ball  13cm  in  diameter,  specific  gravity  7.29 ;  of  a  lead  ball 
17.3cm  in  diameter,  specific  gravity  11.35 ;  of  a  silver  ball 
1.31cm  in  diameter,  specific  gravity  10.47. 

55.  A  slab  of  cast  iron  4  ft.  2£  in.  long,  17  in.  wide,  and 
8£  in.  thick,  specific  gravity  7.31,  is  cast  into  2-lb.  balls. 
If  there  is  a  loss  of  5%  in  melting,  how  many  balls  are 
obtained,  and  what  is  the  diameter  of  each  ? 

56.  How  many  pounds  will  a  ball  of  iron  30  in.  in 
diameter  weigh,  if  the  specific  gravity  of  the  iron  is  7.31  ? 

57.  If  the  specific  gravity  of  ice  is  0.930,  find  the  weight 
and  the  surface  of  each  of  three  spheres  of  ice  whose  diam- 
eters are  lcm,  10cm,  and  lm. 

58.  Find  the  capacity  in  gallons  of  a  round  cistern  13  ft. 
in  diameter  and  9  ft.  deep. 

59.  A  cylinder  is  10  in.  in  diameter  and  12  in.  long. 
Find  the  area  of  each  end,  the  lateral  surface,  the  total  sur- 
face, and  the  contents  in  gallons. 

60.  What  must  be  the  diameter  of  a  cylinder  10  in.  deep 
that  it  may  hold  1  gallon  ? 


MISCELLANEOUS    PROBLEMS.  377 

61.  Find  the  volume  of  a  cylinder  8  in.  in  diameter  and 
11  in.  high. 

62.  Find  the  dimensions  of  three  cylinders  that  have  the 
diameters  equal  to  the  heights,  and  hold  1  gallon,  1  quart, 
and  1  liter,  respectively. 

63.  How  many  cubic  yards  in  a  pyramid  123  ft.  high, 
with  a  square  base  210  ft.  on  a  side  ? 

64.  Find  the  capacity  of  a  cup,  whose  mouth  is  4  in. 
square,  and  whose  sides  are  four  equilateral  triangles. 

65.  The  largest  of  the  Egyptian  pyramids  is  147m  high, 
with  a  base  231m  square.     Find  its  volume  in  cubic  meters. 

66.  The  slant  depth  of  a  conical  cup  is  93mm,  and  the 
diameter  at  the  top  8cm.     What  is  its  capacity  ? 

67.  The  volume  of  a  cone  is  lcbm  ;  its  height  is  equal  to 
the  radius  of  its  base.     Find  the  dimensions  of  the  cone. 

68.  Find  the  capacity  in  pints  of  a  cylinder,  diameter 
1.9375  in.,  height  2.4375  in.;  of  a  cylinder,  diameter  3£  in., 
height  3|  in. ;   of  a  cylinder,  diameter  3}|  in.,  height  5T^  in. 

69.  Find  the  capacity  in  pecks  of  a  cylinder,  diameter 
15.865  in.,  height  12.5  in.;  of  a  cylinder,  diameter  9.25  in., 
height  4.25  in.;  of  a  cylinder,  diameter  18.5  in.,  height  8  in. 

70.  What  must  be  the  diameter  of  a  circle  to  contain 
78.54  sq.  ft.?   to  contain  314.16  sq.  ft.? 

71*  What  must  be  the  diameter  of  a  circle  to  contain 
1  A.?   to  contain  9  A.? 

72.  What  must  be  the  diameter  of  a  circle  to  contain 
lha  ?    to  contain  25ha  ? 

73.  Divide  $1270  into  parts  proportional  to  4^,  5 J,  6f. 

74.  How  much  water  will  a  hemispherical  bowl  hold 
that  is  10  in.  in  diameter  ? 

75.  At  50  cents  a  square  foot,  what  will  it  cost  to  gild 
a  hemispherical  dome  10  ft.  in  diameter  ? 

76.  If  the  moon  is  a  sphere  2170  miles  in  diameter,  how 
many  million  bushels  would  it  hold  if  hollow  ? 


378  MISCELLANEOUS   PROBLEMS. 

77.  If  the  earth  is  7920  miles  in  diameter,  and  the 
air  is  40  miles  deep,  how  many  cubic  miles  of  air  are 
there  ? 

78.  What  is  the  difference  between  2  feet  square  and 
2  square  feet  ?  between  a  foot  square  and  a  square  foot  ? 
between  half  a  foot  square  and  6  in.  square  ? 

79.  Find  the  volume  of  a  frustum  of  a  right  pyramid 
whose  lower  base  is  a  square  3  ft.  on  a  side,  upper  base  a 
square  2  ft.  on  a  side,  and  height  4  ft. 

80.  Find  the  capacity  in  liquid  quarts  of  a  tin  pan  10  in. 
in  diameter  at  the  top,  8  in.  in  diameter  at  the  bottom,  and 
4  in.  deep. 

81.  How  many  hektoliters  will  a  circular  vat  hold  5m  in 
diameter  at  the  top,  4.57m  in  diameter  at  the  bottom,  and 
1.17m  deep  ? 

82.  If  4  cu.  in.  of  iron  weigh  1  lb.  avoirdupois,  what  is 
the  weight  in  grains  of  1  cu.  in.  of  iron  ?  What  is  the 
specific  gravity  of  the  iron  ? 

83.  If  4  cu.  in.  of  iron  weigh  1  lb.,  what  is  the  diameter 
of  a  6-lb.  ball  ?   of  a  32-lb.  ball  ? 

84.  At  £  lb.  to  the  cubic  inch,  what  is  the  weight  of  a 
rectangular  block  of  iron  17.36  in.  by  8.7  in.  by  1.76  in.? 
What  would  be  its  diameter  if  cast  into  a  ball,  if  11%  is 
allowed  for  waste  ? 

85.  At  £  lb.  to  the  cubic  inch,  what  is  the  weight  of  a 
rectangular  block  of  iron  71.4  in.  by  8  J  in.  by  3£  in.? 
What  would  be  its  diameter  if  cast  into  a  ball,  if  11%  is 
allowed  for  waste  ? 

86.  What  is  the  diameter  of  a  cylinder  11  in.  long  that 
will  hold  2  gallons  ? 

87.  What  is  the  diameter  of  a  cylinder  9  in.  long  that 
will  hold  2  gallons  ? 

88.  What  is  the  diameter  of  a  cylinder  30cm  long  that 
will  hold  10  liters  ? 


MISCELLANEOUS   PROBLEMS.  379 

89.  Find  the  circumference  of  a  globe,  if  the  number  of 
square  centimeters  in  its  surface  is  three  times  the  number 
of  cubic  centimeters  in  its  volume. 

90.  Find  the  diameter  of  a  circle,  if  the  number  of  inches 
in  its  circumference  is  equal  to  the  number  of  square  feet 
in  its  area. 

9 1 .  How  many  times  does  a  carriage  wheel  3  ft.  2  in.  in 
diameter  turn  in  going  a  mile  on  a  smooth  road  ? 

92.  A  point  in  the  tire  moves,  while  the  wheel  turns 
once,  just  four  times  the  diameter  of  the  wheel.  How  far 
does  a  spike  head  in  the  tire  travel  while  a  wheel,  3  ft. 
2  in.  in  diameter,  travels  1  mi.? 

93.  An  oil  can  is  formed  of  two  cylinders  connected  by 
a  frustum  of  a  cone.  .  The  upper  cylinder,  or  neck,  is  6cm  in 
diameter,  and  75mm  high  ;  the  lower  cylinder  is  13cm  in 
diameter,  and  153mm  high ;  the  total  length  of  the  can 
is  30cm.     Find  the  capacity  of  the  can  in  liters. 

94.  A  common  tunnel  is  formed  of  a  frustum  of  a  cone 
terminated  with  a  cylinder.  The  height  of  the  frustum  is 
14cm,  and  the  diameters  of  the  two  bases  are  175mm  and 
16m,n,  respectively.  The  cylinder  is  8cm  long.  Find  the 
capacity  of  the  tunnel  in  liters. 

95.  A  pan  in  the  form  of  a  frustum  of  a  cone  is  10cm 
deep,  J.2cm  across  the  bottom,  and  23cm  across  the  top.  Find 
the  capacity  of  the  pan  in  liters. 

96.  Find  the  number  of  square  centimeters  of  sheet  iron 
in  a  stovepipe  4m  long,  26cm  in  diameter,  and  lmm  thick,  if 
the  edges  lap  one  centimeter.  Find  the  weight  of  the  pipe, 
if  the  specific  gravity  of  the  sheet  iron  is  7.8. 

97.  A  steam  boiler  is  formed  of  a  cylinder  terminated 
at  each  end  by  a  hemispherical  cap  of  the  same  diameter. 
The  length  of  the  cylinder  is  3.4m,  interior  diameter  0.8m. 
Find  the  number  of  hektoliters  of  water  required  to  fill 
the  boiler  half  full. 


380  MISCELLANEOUS    PROBLEMS. 

98.  A  spherical  bomb  is  32cm  in  diameter,  and  the  sides 
38mm  thick.  If  the  specific  gravity  of  the  metal  is  7.2, 
what  is  the  weight  of  the  bomb  and  its  capacity  ? 

99.  The  diameters  of  a  lamp  shade  are  25cm  and  7cm,  and 
its  slant  height  is  134mm.  Find  its  curved  surface  in  square 
centimeters. 

100.  A  niche  is  formed  like  a  half -cylinder  surmounted 
by  a  quarter  of  a  sphere.  The  height  of  the  cylinder  is 
1.2m,  the  diameter  0.8m.  Find  the  volume  of  the  niche,  and 
the  area  of  its  interior  surface. 

101.  What  is  the  expense,  at  30  cents  a  square  yard,  of 
painting  the  walls  and  ceiling  of  a  room  22  ft.  6  in.  long, 
13  ft.  6  in.  wide,  and  10  ft.  high  ? 

102.  In  what  time  will  an  empty  cistern  be  filled  by 
three  pipes  whose  diameters  are  £  in.,  £  in.,  and  1  in.,  if 
the  largest  alone  would  fill  it  in  40  min.?  The  rates  of 
flow  are  proportional  to  the  squares  of  the  diameters. 

103.  How  many  gallons  of  water  are  contained  in  a 
length  of  50  yd.  of  a  canal,  if  its  width  at  the  top  is  8  yd. 
and  at  the  bottom  7  yd.,  and  its  depth  5  ft.? 

104.  A  man  who  rows  4  miles  an  hour  in  still  water 
takes  1  hr.  12  min.  to  row  4  miles  up  a  river.  How  long 
will  it  take  him  to  row  down  again  ? 

105.  How  long  must  a  ladder  be  to  reach  a  window 
40  ft.  from  the  ground,  if  the  distance  of  the  foot  of  the 
ladder  from  the  wall  is  9  ft.? 

106.  If  3  oz.  of  gold  15  carats  fine  are  mixed  with  7  oz. 
12  carats  fine,  what  will  be  the  fineness  of  the  compound  ? 
What  must  be  the  fineness  of  11  oz.  that,  when  added  to 
this  compound,  the  whole  may  be  14  carats  fine  ? 

107.  Find  the  surface  of  each  face  of  a  cube  whose 
volume  is  14  cu.  ft.  705.088  cu.  in. 

108.  Determine  the  depth  of  conical  wineglasses  2\  in. 
across  the  top  that  60  of  them  may  hold  a  gallon. 


MISCELLANEOUS    PROBLEMS.  381 

109.  What  must  be  the  length  of  spermaceti  candles  $• 
of  an  inch  in  diameter  that  six  of  them  may  weigh  a  pound, 
if  the  specific  gravity  of  spermaceti  is  0.943  ? 

110.  A  cylinder  10  in.  across  and  10  in.  high  contains 
0.3927  cu.  ft.  of  water.  How  many  shot  0.1  in.  in  diam- 
eter must  be  poured  in  to  raise  the  water  to  the  top  ? 

111.  How  deep  must  a  round  cistern  4  ft.  in  diameter  be 
made  to  be  lined  with  the  same  amount  of  lead  as  a  cubical 
cistern  4  ft.  on  an  edge  ?     Compare  their  capacities. 

112.  The  material  for  lining  a  cubical  cistern  cost  $10. 
Find  the  cost  of  the  material  for  lining  two  similar  cisterns 
which  shall  each  hold  one  half  as  much. 

113.  If  5  excavators  sink  a  circular  shaft  8  ft.  in  diam- 
eter and  125  fathoms  deep  in  100  days  of  10  hr.  each,  how 
many  nights  of  7  hr.  each  will  4  excavators  be  in  sinking 
a  shaft  6  ft.  in  diameter  and  75  fathoms  deep,  if  the  diffi- 
culty of  working  by  night  is  one  seventh  greater  than  by 
day,  and  the  hardness  of  the  ground  in  the  smaller  shaft  is 
to  that  in  the  larger  shaft  as  7  is  to  5  ? 

114.  Find  the  number  of  dry  quarts  a  tub  will  hold  that 
is  22  in.  across  the  top,  20  in.  across  the  bottom,  and  18  in. 
deep. 

115.  Find  the  number  of  dry  quarts  a  cylinder  will 
hold  that  is  28  in.  long  and  has  a  diameter  of  18  in. 

116.  How  high  will  2  quarts  of  milk  stand  in  a  cylindri- 
cal pail  7  in.  in  diameter  ?  How  high  will  2  quarts  of  oats 
stand  in  the  same  pail  ? 

117.  Find  the  capacity  in  gallons  of  a  cylindrical  boiler 
1  ft.  in  diameter  and  4  ft.  10  in.  long;  of  a  cylindrical 
boiler  1  ft.  6  in.  in  diameter  and  3  ft.  6  in.  long;  of  a 
cylindrical  boiler  2  ft.  8  in.  in  diameter  and  5  ft.  6  in.  long. 

118.  Find  the  capacity  of  a  tumbler  3J-  in.  across  the 
bottom,  3-J-  in.  across  the  top,  and  3^-  in.  deep ;  of  a  cylin- 
drical tumbler  3£  in.  in  diameter  and  3£  in.  deep. 


382  MISCELLANEOUS    PROBLEMS. 

686.  Ellipse.  If  a  point  moves  continuously  so  that 
the  sum  of  its  distances  from 
two  fixed  points,  called  the 
foci,  always  remains  the 
same,  the  point  traces  a 
curve  called  an  ellipse. 

Two  common  examples  of  an 
ellipse  are  the  shadow  of  a  circular  plate  and  a  section  of  a  right 
cylinder  not  parallel  to  the  bases. 

687.  To  Find  the  Area  of  an  Ellipse, 

Multiply  the  product  of  its  longest  and  shortest  diameters 
by  0.7854  (i  of  8.1^16). 

119.  Find  the  area  of  an  ellipse  whose  longest  and 
shortest  diameters  are  11  in.  and  8  in.,  respectively. 

120.  The  ends  of  a  rope  100  ft.  long  are  fastened  to 
stakes  placed  80  ft.  apart  on  level  ground.  A  ring,  to 
which  a  kid  is  tied,  plays  freely  on  the  rope.  How  far 
from  a  straight  line  joining  the  stakes  can  the  ring  be  pulled  ? 

121.  If  the  stakes  of  Ex.  J  20  are  placed  25  ft.  apart,  by 
how  many  per  cent  is  the  kid's  pasturage  increased,  pro- 
vided he  can  graze  18  in.  beyond  the  rope  when  stretched  ? 

122.  A  cylindrical  log,  11  in.  in  diameter,  is  sawed  off 
at  such  a  slant  that  the  pieces  are  8  in.  longer  on  the 
longest  than  on  the  shortest  side.  Find  the  diameters  of 
the  ellipse  thus  made,  and  its  area. 

123.  Find  the  area  of  an  ellipse,  if  its  longest  diameter 
is  12  in.  and  its  shortest  diameter  9  in. 

688.  To  Find  the  Capacity  of  Any  Round  Vessel,  like 
a  Cup,  Saucer,  Bowl,  or  Tunnel, 

If  spherical,  the  vessel  holds  two  th  irds  as  much  as  a 
cylinder  of  the  same  diameter  and  depth;  if  conical,  one 
third  as  much;   if  like  a  coffee  cup,  one  half  as  much. 


MISCELLANEOUS    PROBLEMS.  383 

124.  Find  the  number  of  quarts  a  conical  tunnel  will 
hold  if  it  is  9  in.  across  the  top  and  8  in.  deep. 

125.  Find  the  number  of  pints  a  spherical  bowl  will 
hold  if  it  is  5  in.  across  the  top  and  2\  in.  deep. 

126.  Find  the  number  of  pints  a  spherical  bowl  will 
hold  if  it  is  4  in.  across  the  top  and  3£  in.  deep. 

127.  Find  the  capacity  in  pints  of  a  coffee  cup  3  in. 
across  the  top  and  3  in.  deep. 

128.  Find  the  capacity  in  liters  of  a  spherical  wash 
bowl  30cm  in  diameter  and  5cm  deep. 

129.  Find  the  capacity  in  liters  of  the  basin  of  a  foun- 
tain 89cm  in  diameter  and  31 cm  deep. 

130.  Find  the  capacity  in  quarts  of  a  bowl  10  in.  in 
diameter  and  4  in.  deep. 

131.  Find  the  capacity  in  pints  of  a  saucer  6  in.  across 
and  \\  in.  deep  ;    of  a  bowl  7  in.  across  and  3  in.  deep. 

132.  How  many  gallons  will  a  spherical  basin  5  ft.  in 
diameter  and  2  ft.  deep  hold  ? 

133.  How  many  gallons  will  a  spherical  bowl  30  in.  in 
diameter  and  1  ft.  deep  hold  ? 

134.  Find  the  capacity  in  pints  of  a  saucer  5  in.  across 
and  2  in.  deep. 

135.  Find  the  capacity  in  gallons  of  a  paraboloid  (shaped 
like  a^coffee  cup)  boiler  25  in.  across  and  14  in.  deep. 

136.  Find  the  capacity  in  quarts  of  a  conical  funnel 
9  in.  across  and  7  in.  deep. 

689.   To  Find  the  Number  of  Gallons  a  Cask  will  Hold, 

Multiply  by  0.65  the  difference  between  the  Imng  and  head 
diameters  expressed  in  inches,  and  add  the  product  to  the 
head  diameter  for  the  mean  diameter. 

Divide  the  product  of  the  length  of  the  cask  expressed  in 
inches  and  the  square  of  the  mean  diameter  by  294;  ^ie 
quotient  is  the  number  of  gallons  the  cask  will  hold. 


384  MISCELLANEOUS    PROBLEMS. 

137.  Find  the  number  of  gallons  contained  in  a  full  cask 
whose  bung  diameter  is  24  inches,  head  diameter  22  inches, 
and  length  30  inches. 

138.  Find  the  number  of  gallons  contained  in  a  full  cask 
whose  bung  diameter  is  22  inches,  head  diameter  20  inches, 
and  length  28  inches. 

139.  Find  the  number  of  gallons  contained  in  a  full  cask 
whose  bung  diameter  is  20  inches,  head  diameter  18  inches, 
and  length  28  inches. 

690.  Sound  travels  in  still  air  at  32°  F.  1090  ft.  a  sec- 
ond, and  1.1  ft.  a  second  faster  for  every  degree  Fahrenheit 
increase  in  temperature. 

Sound  travels  in  still  air  at  0°  C.  332m  a  second,  and 
60.9cm  a  second  faster  for  every  degree  Centigrade  increase 
in  temperature.    . 

140.  The  flash  of  a  gun  is  seen  7£  sec.  before  the  report 
of  the  gun  is  heard  ;  there  is  no  wind,  and  the  temperature 
is  73°  F.     How  far  off  is  the  gun  ? 

141.  A  meteor  was  seen  to  burst;  the  report  followed 
in  4  min.  17  sec.  What  was  its  distance,  if  the  average 
temperature  of  the  intervening  air  was  50°  F.? 

142.  How  long  will  it  take  for  an  explosion  at  the  equa- 
tor to  be  heard  at  the  antipodes  of  the  place,  if  the  circum- 
ference of  the  earth  at  the  equator  is  reckoned  at  40,000km, 
and  the  average  temperature  at  the  equator  at  23°  C? 

143.  If  an  explosion  at  the  equator  occurs  at  sunset  and 
the  average  temperature  east  of  the  spot  is  22°  C,  and  that 
to  the  west  24°  C,  how  far  from  the  antipodes  will  the 
sound  waves  meet  ? 

144.  How  far  off  is  the  lightning  when  the  thunder  fol- 
lows in  13  sec,  the  temperature  being  76°  F.? 

145.  How  long  would  it  take  sound  to  go  through  a 
whispering  tube  3  mi.  long,  temperature  61°  F.? 


MISCELLANEOUS   PROBLEMS.  385 

146.  Sound  travels  in  iron  about  10£  times  as  fast  as  in 
air.  How  long,  then,  after  seeing  the  blow  of  a  sledge 
hammer  given  on  the  other  end  of  an  iron  pipe  \\  mi.  long, 
may  I  expect  to  hear  the  sound  by  the  iron ;  and  how  long 
after,  to  hear  the  sound  through  the  air  in  the  pipe  j  ther- 
mometer 63°  F.  ? 

147.  Two  gunners  fire  at  each  other  simultaneously  from 
forts  \\  mi.  apart;  the  wind,  at  70° F.,  blows  steadily  from 
one  fort  to  the  other,  at  11  mi.  an  hour.  How  soon  will 
each  hear  the  report  of  the  other's  gun  ?  Suppose  one  ball 
flies  on  the  average  987  ft.  a  second,  the  other  818  ft.  a 
second ;    when  will  each  receive  the  other's  shot  ? 

148.  Sound  travels  in  water  about  4.26  times  as  fast  as 
in  air.  How  many  seconds  sooner  would  the  sound  of  a 
torpedo  exploded  under  water  2  mi.  off  reach  you  by  water 
than  by  air,  at  68°  F.  ? 

691.  In  the  average  state  of  the  atmosphere,  the  distance 
at  which  an  object  is  visible  at  sea  may  be  found  by  the 
following  relation  : 

The  square  of  the  distance  in  miles  is  seven  fourths  the 
height  of  the  object  in  feet. 

The  square  of  the  distance  in  kilometers  is  fifteen  times 
the  height  in  meters. 

Hence,  log  miles  =  0.1215  +  \  log  feet. 

log  kilometers  =  0.5880  +  i  log  meters. 

149.  A  hill  482  ft.  high  is  8  mi.  from  the  shore.  How 
many  miles  out  at  sea  is  it  visible  ? 

150.  A  sailor  at  the  topmast  80  ft.  above  the  sea  can 
just  see  a  sailor  at  the  topmast  of  a  similar  ship.  How 
many  miles  apart  are  the  vessels  ? 

151.  How  far  is  a  mountain  1000m  high  visible  ?  a 
mountain  2000m  high? 


386  MISCELLANEOUS    PROBLEMS. 

152.  If  a  man  stands  on  a  bluff  that  raises  his  eyes  llin 
above  the  sea,  how  far  can  he  see  from  the  shore  ? 

153.  A  sailor  at  sea  is  at  a  distance  of  171km  from  a 
mountain  when  the  top  of  the  mountain  is  just  visible. 
How  high  is  the  mountain  ? 

154.  A  vessel  approaching  Valparaiso  at  daybreak  just 
makes  out  the  peak  of  Aconcagua,  22,427  ft.  high  and  140 
mi.  back  from  the  coast.  How  far  is  the  vessel  from  land 
if  the  eye  of  the  observer  is  30  ft.  above  the  water  ? 

155.  If  Mount  Washington  is  6293  ft.  high  and  76  mi. 
in  an  air  line  from  Cape  Elizabeth,  how  far  out  from  the 
Cape  will  its  peak  be  visible  in  the  ordinary  state  of  the 
atmosphere  ? 

156.  How  many  acres  of  water  can  a  man  see  if  he 
Stands  on  a  raft  with  his  eyes  just  6  ft.  above  the  water, 
and  no  land  is  in  sight  ? 

157.  How  far  would  a  mountain  29,000  ft.  high  be 
visible  ?   one  5000  ft.  high  ?   one  1000  ft.  high  ? 

158.  How  high  must  a  mountain  be  in  order  to  be  visible 
at  sea  level  50  miles  ?   100  miles  ?    150  miles  ? 

159.  What  distance  can  be  seen  from  the  top  of  a 
mountain  4  miles  high? 

692.  Pendulum.  A  body  suspended  by  a  straight  line 
from  a  fixed  point  so  as  to  swing  freely  is  called  a  pendulum. 

693.  The  number  of  vibrations  that  pendulums  make  in 
a  given  time  is  inversely  as  the  square  root  of  their  lengths. 

A  pendulum  that  passes  its  central  point  of  rest  once  every  mean 
solar  second  is  39.138  in.  long. 

160.  Find  the  length  of  a  pendulum  that  beats  half- 
seconds  ;   of  a  pendulum  that  beats  quarter-seconds. 

161.  How  many  centimeters  long  is  a  pendulum  that 
swings  80  times  a  minute  ?  a  pendulum  that  swings  30 
times  a  minute  ? 


MISCELLANEOUS    PROBLEMS.  387 

162.  If  a  cannon  ball  is  suspended  by  a  line  wire  176  ft. 
long  in  the  central  well  of  the  Bunker  Hill  Monument, 
how  many  times  a  minute  will  it  swing  ? 

163.  How  long  is  a  pendulum  that  swings  three  times 
in  two  seconds  ?    that  swings  five  times  in  two  seconds  ? 

694.  A  body  falling  from  rest  in  a  vacuum  falls  16^  ft. 
or  4.903m  in  the  first  second ;  it  then  has  acquired  a  veloc- 
ity of  32£  ft.  or  9.806m. 

A  falling  body  increases  its  velocity  in  proportion  to  the 
time  it  is  falling  ;  and  the  distance  fallen  is  in  proportion  to 
the  square  of  the  number  of  seconds  of  time  it  is  falling. 

Thus,  a  body  falling  from  rest  in  a  vacuum  in  half  a  second  has 
fallen  4^-g-  ft.  or  1.2257m,  and  has  acquired  a  velocity  of  16  r^  ft.  or 
4.903m  per  second  ;  in  3  sec.  it  has  fallen  144|  ft.  or  44.127m,  and  has 
acquired  a  velocity  of  96^  ft.  or  29.418m  per  second. 

The  velocity  of  heavy  bodies  falling  short  distances  in  air  will  not 
be  much  less  than  in  a  vacuum. 

164.  What  velocity  in  meters  a  second  will  a  cannon 
ball  acquire  in  falling  three  quarters  of  a  second  ?  in  fall- 
ing three  and  a  quarter  seconds  ? 

165.  How  long  will  it  take  a  leaden  ball,  rolling  off  a 
table  29  in.  high,  to  reach  the  floor  ? 

16€.  What  velocity  will  a  crowbar  attain  in  falling  end- 
wise from  a  balloon  2000m  high  ?  How  long  will  it  be  in 
coming  down  ? 

167.  What  velocity  will  a  crowbar  attain  in  falling  end- 
wise from  a  balloon  one  mile  and  a  quarter  high  ?  How 
long  will  it  be  coming  down  ? 

168.  How  long  will  it  take  a  ball,  rolling  off  a  table,  to 
drop  lcm  ?    1  in.  ?    10cm  ?    6  in  ? 

169.  If  Carisbrook  Well  is  210  ft.  deep,  how  long  after 
a  pebble  is  dropped  will  it  be  heard  to  strike  the  bottom, 
if  the  velocity  of  sound  is  1120  ft.  a  second  ? 


588  MISCELLANEOUS    PROBLEMS. 

170.  How  long  after  a  pebble  is  dropped  will  it  be  heard 
to  strike  the  bottom  of  a  ventilating  shaft  1600  ft.  deep, 
if  the  temperature  is  68°  F.? 

171.  If  a  rock  dropped  over  a  precipice  strikes  the 
bottom  in  7£  sec,  how  high  is  the  precipice  ? 

172.  How  long  after  a  pebble  dropped  down  a  shaft 
133  ft.  deep  will  it  be  heard  to  strike  the  bottom,  if  the 
temperature  is  59°  F.? 

695.  If  a  plunger  fits  tightly  in  a  small  cylinder,  and  by 
it  water  is  forced  into  a  large  cylinder,  a  plunger  in  the  large 
cylinder  is  lifted  with  a  force  nearly  equal  to  the  product  of 
the  force  with  which  the  little  plunger  is  driven  in  multiplied 
by  the  square  of  the  ratio  of  the  diameters  of  the  two  cylinders. 

173.  Find  the  lifting  power  of  a  hydraulic  press,  the 
plunger  being  lcm  in  diameter  and  driven  with  a  force  of 
100kg,  if  the  lifting  piston  is  lm  in  diameter. 

174.  If  the  plunger  is  £  in.  in  diameter,  and  is  driven 
with  a  force  of  1000  lb.,  how  much  can  it  lift  with  a  lifting 
piston  4  ft.  in  diameter  ? 

175.  If  the  plunger  is  2  in.  in  diameter,  and  is  driven 
with  a  force  of  1000  lb.,  how  much  can  it  lift  with  a  lifting 
piston  2  ft.  in  diameter  ? 

176.  The  water  stands  in  a  fissure  in  a  rock  10m  high 
and  12m  long.  What  pressure  is  exerted  to  split  the  rock 
on  the  lowest  meter's  width  ?  on  the  highest  meter's 
width  ?    in  the  whole  fissure  ? 

N«te.  This  pressure  is  found  by  multiplying  the  surface  upon  one 
md%  by  the  height  of  water  above  the  centre  line,  counting  the  product 
as  Tolume  of  water,  and  then  finding  the  weight  of  this  volume  of 
water.     The  principle  is  precisely  the  same  as  in  the  hydraulic  press. 

177.  A  dam  is  100  ft.  long  and  10  ft.  deep,  and  the 
water  is  just  flowing  over  it.  What  pressure  is  exerted 
•n  the  lowest  two  feet  of  the  dam? 


MISCELLANEOUS    PROBLEMS. 

178.  Water  is  running  2  ft.  over  a  dam  that  is  180  ft. 
long  and  12  ft.  deep.     Find  the  pressure  on  the  dam. 

179.  Water  is  running  9  in.  deep  over  a  dam  that  is 
78  ft.  long  and  8  ft.  deep.     Find  the  pressure  on  the  dam. 

696.  The  velocity  with  which  water  will  flow  out  of  a  hole 
in  the  side  of  a  reservoir  is  nearly  proportional  to  the  square 
root  of  the  depth  of  the  hole  below  the  surface  of  the  water  ; 
and  is  about  82  ft.  a  second  at  the  depth  of  16  ft. 

180.  With  what  velocity  will  water  flow  through  a  hole 
9  ft.  below  the  surface  ? 

181.  With  what  velocity  will  water  leave  a  fountain 
having  free  play,  and  a  head  of  25  ft.?   a  head  of  100  ft.? 

182.  If  a  hole  in  the  side  of  a  cistern  4  ft.  below  the 
surface  of  the  water  is  delivering  10  gal.  an  hour,  how 
many  gallons  would  it  deliver  with  5  ft.  more  head  ? 

183.  If  a  pipe  2  in.  in  diameter,  and  1  ft.  long,  inserted 
in  a  dam,  the  head  of  water  being  kept  constant,  delivers 
4  gallons  of  water  a  minute,  how  many  gallons  a  minute 
may  be  expected  when  another  pipe  of  the  same  length,  but 
2\  in.  in  diameter,  is  substituted  for  the  two-inch  pipe  ? 

184.  If  a  one-inch  pipe,  20  in.  long,  is  substituted  for 
the  two-inch  pipe,  1  ft.  long,  in  Example  183,  and  the  flow 
is  fotmd  to  be  5  pints  a  minute,  what  part  of  the  decrease  of 
flow  is  due  to  the  smaller  area  of  the  orifice,  and  what  part 
to  the  increased  friction  on  the  sides  of  the  longer  pipe  ? 

697.  The  quantity  of  water  issuing  from  a  hole  is  in  pro- 
portion to  the  square  root  of  the  head  ;  and  the  velocity  is  in 
proportion  to  the  square  root  of  the  head. 

The  work  which  the  water  can  do  is  in  proportion  to  the 
quantity  multiplied  by  the  square  of  the  velocity  ;   that  is, 

The  work  is  in  proportion  to  the  square  root  of  the  cube  of 
the  head. 


390  MISCELLANEOUS    PROBLEMS. 

185.  A  miller  is  using  water  flowing  through  the  gate- 
way under  4  ft.  head.  How  much  more  work  could  he  do 
if  the  head  was  raised  to  9  ft.?  how  much  more  if  the  head 
was  raised  to  25  ft.? 

698.  Work  is  the  act  of  changing  the  position  of  a  body 
by  overcoming  the  resistance  to  the  change. 

699.  Units  of  Work.  The  unit  of  work  is  the  work 
done  in  raising  a  weight  of  one  pound  through  a  distance 
of  one  foot.     This  unit  is  called  a  foot-pound. 

The  corresponding  metric  unit  is  the  kiloyram-meter. 

700.  Horse  Power.  The  rate  at  which  work  is  done  is 
called  power.  A  horse  power  is  the  power  to  do  33,000  foot- 
pounds of  work  per  minute,  or  550  foot-pounds  per  second. 

186.  A  cross  section  of  a  stream  of  water  is  a  rectangle 
6  ft.  by  2£  ft. ;  the  velocity  is  40  ft.  per  minute.  There  is 
a  fall  of  10  ft.  where  a  water  wheel  is  erected  that  utilizes 
70%  of  the  work.     Find  the  horse  power  of  the  wheel. 

187.  Find  the  horse  power  of  the  wheel  of  Example  186, 
if  the  fall  of  the  water  is  14  ft. 

188.  A  cross  section  of  a  stream  of  water  is  a  rectangle 
5  ft.  by  4  ft.;  the  velocity  is  50  ft.  per  minute.  There  is 
a  fall  of  12  ft.  where  a  water  wheel  is  erected  that  utilizes 
65%  of  the  work.     Find  the  horse  power  of  the  wheel. 

189.  Find  the  horse  power  of  the  wheel  of  Example  188, 
if  the  fall  of  the  water  is  16  ft. 

190.  A  cross  section  of  a  stream  of  water  is  a  trapezoid 
whose  altitude  is  3£  ft.,  and  parallel  sides  6  ft.  and  5  ft., 
respectively  ;  the  velocity  is  150  ft.  per  minute.  There  is 
a  fall  of  9  ft.  where  a  water  wheel  is  erected  that  utilizes 
75%  of  the  work.     Find  the  horse  power  of  the  wheel. 


MISCELLANEOUS    PROBLEMS.  391 

701.  When  a  body  is  moving  in  a  circle,  the  centrifugal 
force  is  about  1.227  of  the  continued  product  of  the  weight  of 
the  body,  the  number  of  feet  in  the  radius  of  the  circle,  and 
the  square  of  the  number  of  revolutions  in  a  second. 

Thus,  a  body  going  round  a  circle  of  5  ft.  radius  once  a  minute 

presses  away  from  the  centre  with  a  force  equal  to  1.227  X  5  X  — 

of  the  weight  of  the  body. 

Note.  When  the  radius  is  measured  in  meters,  the  multiplier 
4.025  must  be  used  in  place  of  1.227. 

191.  If  a  top  3  in.  in  diameter  is  making  200  revolu- 
tions a  second,  with  what  force  does  the  outer  layer  pull 
away  from  the  centre  ? 

192.  If  a  sling  30  in.  long  contains  a  stone  that  weighs 
-J-  lb.,  and  is  whirled  round  80  times  a  minute,  what  is  the 
force  pulling  on  the  string  ? 

193.  With  what  force  does  a  locomotive  that  weighs  60 
tons  running  30  mi.  an  hour,  on  a  curve  of  800  ft.  radius, 
bear  against  the  outer  rail  ?  If  the  locomotive  is  running 
60  mi.  an  hour,  with  what  force  does  it  bear  on  the  outer  rail  ? 

194.  If  washed  wool  is  put  wet  into  a  wire  basket  1.2m 
in  diameter,  and  the  basket  is  set  to  spinning  at  the  rate  of 
180  revolutions  a  second,  with  what  force  is  water  wrung 
out  of  the  wool  ? 

195.  If  steel  pens  are  revolved  in  a  basket  32cm  in  diam- 
eter, 17  revolutions  a  second,  with  what  force  is  the  oil 
drained  from  them  ? 

196.  The  top  of  a  wheel  is  at  each  instant  moving  with 
twice  the  velocity  of  the  carriage,  and  is  moving  in  a  curve 
whose  centre,  at  the  instant,  is  as  far  below  ground  as  the 
point  is  above  ground.  What,  then,  is  the  force  exerted 
to  separate  the  mud  from  the  top  of  a  wheel  3  ft.  2  in.  in 
diameter,  when  the  carriage  is  moving  at  the  rate  of  10 
miles  an  hour? 


392 


MISCELLANEOUS    PROBLEMS. 


702.  When  a  chain  of  uniform  thickness  hangs  from 
two  points  not  in  the  same  vertical  line,  it  hangs  in  a  curve 
called  a  common  catenary. 

The  length  of  the  chain  from  the'  lowest  point  to  any  point  selected 
may  be  called  half-chain.  The  height  of  the  point  selected  above  the 
lowest  point  is  called  sag.  The  horizontal  distance  of  the  point 
selected  from  the  lowest  point  is  called  half-span.  The  horizontal 
force  with  which  the  point  selected  is  drawn  inward  is  called  tension. 
The  radius  of  the  circle  which  will  fit  the  curve  at  its  lowest  point  is 
called  radius.  The  straight  line  which  touches  the  curve  at  the  point 
selected  is  called  tangent. 

703.  1.  The  tension  is  equal  to  the  weight  of  a  piece  of 
the  same  chain  as  long  as  the  radius. 

2.  The  radius  is  equal  to  the  sum  of  the  half-chain  and 
sag  multiplied  by  the  difference  of  the  half-chain  and  sag, 
and  divided  by  twice  the  sag. 

3.  The  log  of  half-span  is  equal  to  the  log  of  sum  of  half- 
chain  and  sag  plus  log  of  their  difference,  plus  log  of  the  dif- 
ference of  these  two  logarithms  plus  colog  of  sag  plus  0.0612. 

4.  Radius  divided  by  half-chain  measures  the  "batter"  at 
the  point  selected ;  that  is,  measures  the  horizontal  falling 
back  for  every  unit  of  vertical  ascent  in  a  straight  line 
tangent  at  that  point. 

197.  How  strong  a  horizontal  pull  on  a  chain,  weighing 
half  a  pound  to  the  yard,  is  required  to  make  the  lowest 
part  curve  with  an  18-in.  radius  ?    with  a  6-ft.  radius  ? 


MISCELLANEOUS   PROBLEMS.  393 

198.  A  f-in.  rope,  weighing  £  lb.  to  the  yard,  is  fastened 
at  one  end  to  a  staple,  and  near  the  other  end,  on  the  same 
level,  runs  over  a  pulley,  and  has  a  25-lb.  weight  hung  to  it. 
What  is  the  radius  of  its  curvature  at  the  middle  ? 

199.  A  shower  wets  the  rope  of  Example  198,  and  in- 
creases its  weight  40  °f0  ;  what  does  its  radius  now  become  ? 

200.  A  steam  tug,  in  attempting  to  move  a  ship,  straight- 
ened the  hawser  until  the  radius  of  the  lowest  point  was 
1980  ft.  The  rope  was  wet,  and  weighed  3£  lb.  to  the 
yard.     With  what  force  was  it  stretched  ? 

201.  A  chain  31  ft.  long  hangs  between  points  on  a 
level,  and  sags  4  ft.  What  is  the  radius  at  the  lowest 
point  ? 

202.  The  whole  chain,  in  Example  201,  weighs  18  lb. 
What  is  the  horizontal  tension  ?  What  is  the  distance 
between  the  points  ?  What  is  the  slant,  or  batter,  of  the 
end  of  the  chain  ? 

203.  A  chain  weighing  lkg  to  the  meter  is  suspended 
from  points  on  a  level  ;  the  length  of  chain  is  31m,  and  it 
sags  1.3m.  Find  all  the  conditions,  and  find  how  much  it 
falls  below  a  level  at  10cra  from  each  end. 

204.  A  chain  100m  long,  weighing  14  oz.  to  the  foot, 
is  suspended  from  points  on  a  level  80m  apart,  and  sags 
26.5£>.m.  What  is  the  radius  at  the  middle,  the  batter  at 
the  ends,  and  the  horizontal  tension  ? 

205.  If  the  chain  of  Example  204  is  shortened  5m,  the 
sag  is  decreased  4m.     What  is  the  radius  and  tension  ? 

704.  A  lever  is  a  rigid  bar  that  can  be  moved  about  a 
fixed  axis  called  the  fulcrum. 

The  perpendicular  from  the  fulcrum  to  the  line  in  which 
the  power  acts  is  the  power  arm  of  a  lever. 

The  perpendicular  from  the  fulcrum  to  the  line  in  which 
the  weight  acts  is  the  weight  arm  of  a  lever. 


394  MISCELLANEOUS    PROBLEMS. 

705.  The  ratio  of  the  power  to  the  weight  raised  by  a 
lever  is  equal  to  the  ratio  of  the  weight  arm  to  the  power  arm. 

206.  How  heavy  a  rock  placed  6  in.  from  the  fulcrum 
can  a  man,  who  weighs  180  lb.,  raise  with  a  crowbar  5  ft. 
6  in. long  ? 

207.  Two  weights  of  30  lb.  and  20  lb.,  respectively,  at 
the  ends  of  a  horizontal  lever  5  ft.  long  balance.  Find 
how  far  and  in  which  direction  the  fulcrum  must  be  moved 
for  the  weights  to  balance  when  each  is  increased  by  5  lb. 

208.  A  man  who  weighs  160  lb.,  wishing  to  raise  a  rock, 
leans  with  his  whole  weight  on  a  horizontal  crowbar  5  ft. 
long,  which  is  propped  at  the  distance  of  4  in.  from  the 
end  in  contact  with  the  rock.  Find  the  force  he  exerts  on 
the  rock,  and  the  pressure  the  prop  has  to  sustain,  if  the 
weight  of  the  crowbar  is  not  reckoned. 

209.  A  child  weighing  56  lb.  is  seated  at  one  end  of  a 
plank  16  ft.  long,  and  a  child  weighing  72  lb.  is  at  the 
other  end.  Find  the  distance  of  each  child  from  the  ful- 
crum when  the  plank  is  used  for  a  seesaw. 

210.  In  a  pair  of  nutcrackers  if  the  nut  is  placed  at  a 
distance  of  1  in.  from  the  hinge,  and  the  hand  presses  at 
a  distance  of  8  in.  from  the  hinge,  find  the  pressure  upon 
the  nut  for  every  ounce  of  pressure  exerted  by  the  hand. 

211.  A  body  is  weighed  in  both  arms  of  a  false  balance, 
and  its  apparent  weights  are  2.56  lb.  and  2.25  lb.  Find 
its  true  weight. 

212.  In  a  steelyard  the  weight  of  the  beam  is  15  lb., 
and  the  distance  of  its  centre  of  gravity  from  the  fulcrum 
is  3  in.  Find  the  distance  from  the  fulcrum  a  weight  of 
6  lb.  must  be  placed  to  balance  the  beam. 

706.  With  the  Wheel  and  Axle,  the  ratio  of  the  power  to 
the  weight  to  be  raised  is  equal  to  the  ratio  of  the  radius  of 
the  axle  to  the  radius  of  the  wheel. 


MISCELLANEOUS    PROBLEMS.  395 

213.  A  cask  weighing  160kg  is  attached  to  a  rope  wound 
on  an  axle  19cm  in  diameter  ;  at  one  end  of  the  axle  is  a 
wheel  175cm  in  diameter.  With  what  force  must  a  man  pull 
down  on  a  rope  passing  over  the  wheel  to  raise  the  cask? 

214.  A.  rope  passes  over  a  single  pulley.  How  much 
force  is  required  to  raise  180  lb.  attached  to  one  end  of  a 
rope  if  1  %  of  the  force  is  required  to  overcome  friction  ? 

215.  If  the  radius  of  the  wheel  is  four  times  that  of  the 
axle,  and  the  string  round  the  wheel  can  support  a  weight 
of  50  lb.  only,  find  the  greatest  weight  that  can  be  lifted. 

216.  Find  the  ratio  of  the  radii  of  a  wheel  and  axle  that 
a  force  of  100  lb.  may  just  support  a  weight  of  1  ton. 

217.  The  radius  of  a  wheel  is  80cm  and  the  radius  of  the 
axle  is  12cm.  What  weight  can  be  supported  by  a  force  of 
30kg  ?     Find  the  work  done  if  the  weight  is  raised  60cm. 

707.  With  the  Screw,  the  ratio  of  the  power  to  the  weight 
is  equal  to  the  ratio  of  the  distance  between  two  consecutive 
threads  to  the  circumference  described  by  the  end  of  the 
power  arm. 

218.  The  power  arm  of  a  screw  is  16  in.  long,  and  by 
one  turn  of  the  screw  the  head  advances  one  eighth  of  an 
inch.     If  the  power  is  3  lb.,  find  the  weight  lifted. 

219.  In  a  screw  used  to  raise  a  load  of  10  tons,  the 
power  is  50  lb.,  acting  by  an  arm  4  ft.  long.  Find  the 
distance  between  two  consecutive  threads. 

220.  The  lever  of  a  screw  is  1  ft.  9  in.  long,  and  the 
power  applied  at  the  end  is  100  lb.  What  must  be  the 
distance  between  the  threads  that  a  pressure  of  5000  lb. 
may  act  on  the  press  board  ? 

221.  The  lever  of  a  screw  is  3  ft.  6  in.  long,  and  the 
distance  between  the  threads  is  \  in.  What  power  must 
be  applied  at  the  end  of  the  lever  to  produce  a  pressure  of 
10  tons  on  the  press  board  ? 


396  MISCELLANEOUS    PROBLEMS. 

708.  Chemists  employ  initial  letters  to  signify  fixed 
proportional  quantities.  Multiples  of  these  are  indicated 
by  small  figures  placed  after  them ;  and  multiples  of  com- 
pounds are  indicated  by  figures  placed  before  them. 

Thus,  if  the  gram  is  the  unit,  H  signifies  Is  of  hydrogen,  and  O 
signifies  16s  of  oxygen;  then  H20  means  18s  of  water,  and  2  ll2o 
means  36s  of  water  ;  while  H2  +  O  would  mean  2s  of  hydrogen  and 
16s  of  oxygen  mixed,  but  not  chemically  united.  An  electric  spark 
passed  through  H2  +  O  would  produce  H20,  with  an  explosion. 

709.  The  chemical  symbols  and  numerical  proportions 
of  a  few  common  elements  are  as  follows  : 

Element.  Symbol.  Numerical  Equivalent. 

Hydrogen H 1 

Oxygen O 16 

Sulphur S 32 

Iron  (ferrum)    .     .     .     .  Fe 66 

Calcium Ca 40 

Sodium  (natron)    .     .     .  Na 23 

Carbon C 12 

Boron B 11 

222.  What  per  cent  of  water  is  oxygen  ?  what  per  cent 
hydrogen  ? 

223.  What  per  cent  of  quicklime,  CaO,  is  oxygen  ? 

224.  What  per  cent  of  water  in  slacked  lime,  Ca02H2  ? 

225.  What  per  cent  of  pure  marble,  CaC03,  is  oxygen  ? 

226.  What  per  cent  of  gypsum,  called  plaster  of  Paris, 
CaS04  +  2  H20,  is  sulphur  ? 

227.  What  per  cent  of  washing  soda,  Na-jCC^  +  10  H20, 
is  carbon  ? 

228.  In  118  lb.  of  Glauber  salts,  Na^SO,  + 10  H20,  how 
many  ounces  of  sulphur  ? 

229.  How  many  ounces  of  soda,  Na20  +  H20,  in  7  lb. 
of  borax,  Na^O,  + 10  H20  ? 

230.  What  per  cent  of  pure  alcohol,  C2H60,  is  carbon? 
What  per  cent  of  pure  white  marble,  CaC08,  is  carbon  ? 


MISCELLANEOUS    PROBLEMS.  397 

231.  What  per  cent  of  pure  acetic  acid  (the  acid  of 
vinegar)  is  carbon,  the  formula  being  C2H402? 

232.  How  much  acetic  acid  can  be  obtained  from  12  lb. 
of  alcohol,  C2H60,  if  there  is  no  waste  ? 

233.  How  many  grains  of  carbon  in  1  oz.  avoirdupois  of 
oxalic  acid,  C2H204  +  2  H20  ? 

234.  How  many  milligrams  of  carbon  in  3g  of  tartaric 
acid,  C4H606? 

235.  How  many  kilograms  of  carbon  in  95kg  of  white 
sugar,  C12H220ii  ? 

236.  The  formula  of  camphor  is  Ci0H16O.  How  many 
grams  of  carbon  in  14kg  of  camphor  ? 

237.  In  20kg  of  oil  of  vitriol,  H2S04,  how  many  grams  of 
sulphur  ? 

238.  What  per  cent  of  oil  of  vitriol  is  water  ?  what  per 
cent  sulphuric  acid,  S03  ? 

239.  In  3.5g  of  black  oxide  of  iron,  FeO,  how  many  mil- 
ligrams of  jron  ? 

240.  Eed  iron-rust  consists  of  70%  iron  and  30% 
oxygen.      Find  its  formula. 

241.  The  choking  vapor  of  burning  sulphur  is  sulphur 
and  oxygen  in  equal  parts.     Find  its  formula. 

242.  Copperas  is  28.9%  sulphuric  acid,  25.7%  oxide  of 
iron,j45.4%  water.     Find  its  formula. 

Solution.  Water  being  18,  oxide  of  iron  72,  and  sulphuric  acid 
80,  first  seek  multiples  of  72  and  80,  in  the  ratio  of  25.7  to  28.9 ;  that 
is,  of  0.8892  to  1.  But  72  and  80  are  in  almost  exactly  that  ratio. 
This  gives  FeS04  +  water ;  and  it  remains  to  find  a  multiple  of  18 
which  is  to  152  as  45.4  is  to  54.6;  that  is,  which  is  0.8315  of  152,  or 
126.4.  But  7  X  18  =  126;  and  the  addition  of  seveh  parts  of  water 
gives  FeS04  +  7  H20. 

243.  Spirits  of  turpentine  is  11.76%  hydrogen  and 
88.24%  carbon.  Find  its  formula.  What  per  cent  of 
oxygen  combined  with  spirits  of  turpentine  are  required 
to  make  camphor,  C10H16O? 


398  MISCELLANEOUS    PROBLEMS. 

710.  Ohm  and  Ampere.  The  unit  of  the  resistance  of 
the  conductor  to  an  electrical  current  is  an  ohm.  The  unit 
of  the  strength  of  an  electrical  current  through  a  conductor 
is  an  ampere.     These  units  have  been  arbitrarily  chosen. 

Note.  The  strength  of  the  electrical  current  for  an  ordinary  arc 
lamp  is  about  10  amperes. 

711.  A  volt  is  the  force  required  to  send  an  electrical 
current  of  one  ampere  through  a  conductor  that  offers  a 
resistance  of  one  ohm. 

712.  The  resistance  of  the  conductor  to  an  electrical  cur- 
rent is  directly  proportional  to  the  length  of  the  conductor, 
and  inversely  proportional  to  the  area  of  its  cross  section. 

713.  The  strength  of  an  electrical  current  in  amperes  is 
equal  to  the  force  in  volts  divided  by  the  resistance  in  oh /us. 

244.  If  the  resistance  of  1  mile  of  wire  2mm  in  diameter 
is  4.72  ohms,  what  is  the  resistance  of  3  miles  of  wire  of 
the  same  material  3mm  in  diameter  ? 

245.  What  length  of  copper  wire  lmm  in  diameter  has 
the  same  resistance  as  720m  of  copper  wire  4rani  in  diameter  ? 

246.  The  conductivity  of  iron  is  |  that  of  copper.  If 
the  resistance  of  a  copper  wire  1  mile  long  and  £  in.  in 
diameter  is  6.8  ohms,  what  is  the  resistance  of  an  iron 
wire  ^  in.  in  diameter  and  5  miles  long? 

247.  If  50  volts  force  54.8  amperes  of  electrical  current 
through  a  lamp,  what  is  the  resistance? 

248.  If  the  resistance  of  an  electric  lamp  is  2.8  ohms 
when  a  current  of  10  amperes  is  passing  through  it,  what 
is  the  voltage  ? 

249.  Five  arc  lamps  on  a  circuit  have  each  a  resistance 
of  2.35  ohms.  The  resistance  of  the  wires  is  1.2  ohms  and 
of  the  dynamo  is  0.75  ohm.  What  voltage  is  required  to 
send  a  current  of  15  amperes  through  the  circuit  ? 


TABLES. 


399 


Table  of  Specific   Gravities. 


NAME  OF  SUBSTANCE. 


SPECIFIC  GRAVITY 


WEIGHT    OF    CUBIC 
FOOT  IN  POUNDS. 


Acid,  acetic,  strongest    . 

11      nitric,         " 

11      sulphuric,  " 

Air 

Alcohol,  pure  .... 
11  of  commerce  . 
Aluminum,  lightest  .  . 
Brass,  average  .... 
Brick,  common      .     .     . 

"      pressed   .... 

Cedar,  dry 

Cherry,  dry,  average  .  . 
Coal,  bituminous,  average 

"     anthracite,    .     .     . 
Copper,  cast      .... 

Cork 

Glass,  average  .... 
Gold,  pure,  cast  .  .  . 
Granite,  average  .  .  . 
Gypsum,  heaviest .     .     . 

Hydrogen 

Ice 

Iron,  cast,  average     .     . 

"     wrought,  average  . 
Lead,  cast 

"      white 

Lignum  vitse      .... 

Lime 

Marble,  average  .  .  . 
Mercury  ...... 

Milk^ 

Nitrogen 

Oil,  linseed  ..... 
Oak,  dry  white      .     .     . 

Oxygen 

Pine,  dry  white  .  .  . 
Platinum,  hammered 

Salt 

Sand,  average  .... 
Silver,  hammered  .  .  . 
Slate,  lightest    .... 

Steel    

Tin,  cast 

Water,  sea 

Zinc 


1.062 

1.583 

1.841 

0.001292 

0.792 

0.834 

2.560 

7.611 

2.000 

2.400 
.350  to  0.600 

0.715 

1.250 

1.500 

8.788 

0.240 

2.760 
19.258 

2.720 

2.288 

0.0000893 

0.930 

7.150 

7.770 
11.350 

7.235 

1.333 

0.804 

2.720 
13.580 

1.032 

0.001250 

0.940 

0.830 

0.001429 

0.400 
21.841 

2.130 

1.650 
10.500 

2.110 

7.816 

7.291 

1.026 

7.190 


66.4 
'  98.94 
115. 
0.0808 

49.43 

52.1 
160. 
475.7 
125. 
150. 
22  to  37.5 

44.7 

78.1 

93.75 
549.3 

15. 

172.5 

1204. 

170. 

143. 

0.00558 

68.1 
447. 
486. 
709.4 
452. 

83.3 

50.25 
170. 
849. 

64.5 
0.0781 

58.8 

51.9 
0.0893 

25. 
1365. 
133. 
103. 
656.2 
132. 
488.6 
456. 

64.13 
449.4 


400 


TABLES. 


Table  of  Selected  Constants. 


Diagonal  of  square  in  terms  of  side 
Side  in  terms  of  diagonal     .     .     . 
Cube  root  of  2   ......     . 

Cube  root  of  £ 

Square  root  of  20 

Square  root  of  3 

Square  root  of  £ 

Square  root  of  30 

Square  root  of  5 

Extreme  and  mean  ratio      .     .     . 
Extreme  and  mean  ratio      .     .     . 


NUMBER.        LOGARITHM. 


1.4142136 
0.7071068 
1.2599210 
0.7937002 
4.4721359 
1.7320508 
0.5773503 
5.4772256 
2.2360680 
0.3819660 
0.6180340 


0.1505150 
9.8494850 
0.1003433 
9.8996567 
0.6505150 
0.2385606 
9.7614394 
0.7385606 
0.3994850 
9.5820247 
9.7910124 


Circumference  in  terms  of  diameter 
Diameter  in  terms  of  circumference 
Circle  in  terms  of  diameter  square 
Square  in  terms  of  the  circle    .     . 
Sphere  in  terms  of  diameter  cube 
Cube  in  terms  of  sphere  .... 


3.1415927 
0.3183099 
0.7853982 
1.2732395 
0.5235988 
1.9091406 


0.4971499 
9.5028501 
9.8950899 
0.1049101 
9.7189986 
0.2810014 


One  meter  in  feet  .  .  . 
One  foot  in  meters  .  . 
One  meter  in  yards  .  . 
One  yard  in  meters  .  . 
One  centimeter  in  inches 
One  inch  in  centimeters  . 
One  mile  in  kilometers  . 
One  kilometer  in  miles    . 


3.2808693 
0.3047973 
1.0936231 
0.9143918 
0.3937043 
2.5399772 
1.6093295 
0.6213768 


0.5159889 
9.4840111 
0.0388677 
9.9611323 
9.5951702 
0.4048298 
0.2066450 
9.7933550 


One  square  foot  in  square  meters 
One  square  meter  in  square  feet  . 
One  sq.  inch  in  square  centimeters 
One  square  centimeter  in  sq.  inches 

One  acre  in  hektars 

One  hektar  in  acres 


0.0929014 
10.7641036 

6.4514847 
0.15500308 

0.4046784 

2.4710982 


8.9680221 
1.0319779 
0.8096596 
9.1903404 
9.6071100 
0.3928900 


One  cubic  foot  in  cubic  centimeters 
One  cubic  meter  in  cubic  feet  .     . 
One  U.S.  bushel  in  cubic  inches    . 
One  U.S.  bushel  in  cu.  centimeters 
One  cubic  foot  in  U.S.  bushels 
One  U.S.  gallon  in  liters      .     .     . 
One  liter  in  U.S.  gallons      .     .     . 


28,316.085 

35.315017 

2150.42 

35,238.117 

0.8035640 

3.7853103 

0.2641791 


4.4520332 
1.5479668 
3.3325232 
4.5470127 
9.9050204 
0.5781015 
9.4218985 


One  gram  in  grains     . 
One  grain  in  grams     . 
One  pound  in  kilograms 
One  kilogram  in  pounds 


15.4323487 
0.0647990 
0.4535927 
2.2046212 


1.1884320 
8.8115680 
9.6566660 
0.3433340 


VB   17443 


M306073 


W4  £3 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


